Blackholes,gravitationalwavesandfundamental physics: aroadmap · Blackholes,gravitationalwavesandfundamentalphysics: aroadmap 3 Abstract. The grand challenges of contemporary fundamental

Black holes, gravitational waves and fundamentalphysics: a roadmap

Leor Barack1, Vitor Cardoso2,3, Samaya Nissanke4,5,6, ThomasP. Sotiriou7,8 (editors)

Abbas Askar9,10, Krzysztof Belczynski9, Gianfranco Bertone5,Edi Bon11,12, Diego Blas13, Richard Brito14, Tomasz Bulik15,Clare Burrage8, Christian T. Byrnes16, Chiara Caprini17,Masha Chernyakova18,19, Piotr Chruściel20,21, Monica Colpi22,23,Valeria Ferrari24, Daniele Gaggero5, Jonathan Gair25, JuanGarcía-Bellido26, S. F. Hassan27, Lavinia Heisenberg28, MartinHendry29, Ik Siong Heng29, Carlos Herdeiro30, TanjaHinderer4,14, Assaf Horesh31, Bradley J. Kavanagh5, BenceKocsis32, Michael Kramer33,34, Alexandre Le Tiec35, ChiaraMingarelli36, Germano Nardini37a,37b, Gijs Nelemans4,6 CarlosPalenzuela38, Paolo Pani24, Albino Perego39,40, Edward K.Porter17, Elena M. Rossi41, Patricia Schmidt4, AlbertoSesana42, Ulrich Sperhake43,44, Antonio Stamerra45,46, Leo C.Stein43, Nicola Tamanini14, Thomas M. Tauris33,47, L. ArturoUrena-López48, Frederic Vincent49, Marta Volonteri50, BarryWardell51, Norbert Wex33, Kent Yagi52 (Section coordinators)

Tiziano Abdelsalhin24, Miguel Ángel Aloy53, PauAmaro-Seoane54,55,56, Lorenzo Annulli2, Manuel Arca-Sedda57,Ibrahima Bah58, Enrico Barausse50, Elvis Barakovic59, RobertBenkel7, Charles L. Bennett58, Laura Bernard2, SebastianoBernuzzi60, Christopher P. L. Berry42, Emanuele Berti58,61,Miguel Bezares38, Jose Juan Blanco-Pillado62, Jose LuisBlázquez-Salcedo63, Matteo Bonetti64,23, Mateja Bošković2,65,Zeljka Bosnjak66, Katja Bricman67, Bernd Brügmann60, PedroR. Capelo68, Sante Carloni2, Pablo Cerdá-Durán53, ChristosCharmousis69, Sylvain Chaty70, Aurora Clerici67, AndrewCoates71, Marta Colleoni38, Lucas G. Collodel63, GeoffreyCompère72, William Cook44, Isabel Cordero-Carrión73, MiguelCorreia2, Álvaro de la Cruz-Dombriz74, Viktor G. Czinner2,75,Kyriakos Destounis2, Kostas Dialektopoulos76,77, Daniela

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Black holes, gravitational waves and fundamental physics: a roadmap 2

Doneva71,78 Massimo Dotti22,23, Amelia Drew44, ChristopherEckner67, James Edholm79, Roberto Emparan80,81, RecaiErdem82, Miguel Ferreira2, Pedro G. Ferreira83, AndrewFinch84, Jose A. Font53,85, Nicola Franchini7, Kwinten Fransen86,Dmitry Gal’tsov87,88, Apratim Ganguly89, Davide Gerosa43,Kostas Glampedakis90, Andreja Gomboc67, Ariel Goobar27,Leonardo Gualtieri24, Eduardo Guendelman91, FrancescoHaardt92, Troels Harmark93, Filip Hejda2, Thomas Hertog86,Seth Hopper94, Sascha Husa38, Nada Ihanec67, Taishi Ikeda2,Amruta Jaodand95,96, Philippe Jetzer97, XiscoJimenez-Forteza24,77, Marc Kamionkowski58, David E. Kaplan58,Stelios Kazantzidis98, Masashi Kimura2, Shiho Kobayashi99,Kostas Kokkotas71, Julian Krolik58, Jutta Kunz63, ClausLämmerzahl63,100, Paul Lasky101,102, José P. S. Lemos2, JacksonLevi Said84, Stefano Liberati103,104, Jorge Lopes2, RaimonLuna81, Yin-Zhe Ma105,106,107, Elisa Maggio108, AlbertoMangiagli22,23, Marina Martinez Montero86, Andrea Maselli2,Lucio Mayer68, Anupam Mazumdar109, ChristopherMessenger29, Brice Ménard58, Masato Minamitsuji2,Christopher J. Moore2, David Mota110, Sourabh Nampalliwar71

Andrea Nerozzi2, David Nichols4, Emil Nissimov111, MartinObergaulinger53, Niels A. Obers93, Roberto Oliveri112, GeorgePappas24, Vedad Pasic113, Hiranya Peiris27, TanjaPetrushevska67, Denis Pollney89, Geraint Pratten38, NemanjaRakic114,115, Istvan Racz116,117, Miren Radia44, Fethi M.Ramazanoğlu118, Antoni Ramos-Buades38, Guilherme Raposo24,Marek Rogatko119, Roxana Rosca-Mead44, Dorota Rosinska120,Stephan Rosswog27, Ester Ruiz-Morales121, MairiSakellariadou13, Nicolás Sanchis-Gual53, Om Sharan Salafia122,Anuradha Samajdar6, Alicia Sintes38, Majda Smole123, CarlosSopuerta124,125, Rafael Souza-Lima68, Marko Stalevski11,Nikolaos Stergioulas126, Chris Stevens89, Tomas Tamfal68,Alejandro Torres-Forné53, Sergey Tsygankov127, Kıvanç İ.Ünlütürk118, Rosa Valiante128 Maarten van de Meent14 JoséVelhinho129, Yosef Verbin130, Bert Vercnocke86, DanieleVernieri2, Rodrigo Vicente2, Vincenzo Vitagliano131, AmandaWeltman74, Bernard Whiting132, Andrew Williamson4, HelviWitek13, Aneta Wojnar119, Kadri Yakut133, Haopeng Yan93,Stoycho Yazadjiev134, Gabrijela Zaharijas67, Miguel Zilhão2


Abstract. The grand challenges of contemporary fundamental physics—darkmatter, dark energy, vacuum energy, inflation and early universe cosmology,singularities and the hierarchy problem—all involve gravity as a key component. Andof all gravitational phenomena, black holes stand out in their elegant simplicity, whileharbouring some of the most remarkable predictions of General Relativity: eventhorizons, singularities and ergoregions.

The hitherto invisible landscape of the gravitational Universe is being unveiledbefore our eyes: the historical direct detection of gravitational waves by theLIGO-Virgo collaboration marks the dawn of a new era of scientific exploration.Gravitational-wave astronomy will allow us to test models of black hole formation,growth and evolution, as well as models of gravitational-wave generation andpropagation. It will provide evidence for event horizons and ergoregions, test the theoryof General Relativity itself, and may reveal the existence of new fundamental fields.The synthesis of these results has the potential to radically reshape our understandingof the cosmos and of the laws of Nature.

The purpose of this work is to present a concise, yet comprehensive overviewof the state of the art in the relevant fields of research, summarize important openproblems, and lay out a roadmap for future progress. This write-up is an initiativetaken within the framework of the European Action on “Black holes, Gravitationalwaves and Fundamental Physics”.


Glossary

Here we provide an overview of the acronyms used throughout this paper and alsoin common use in the literature.

BBH Binary black holeBH Black holeBNS Binary neutron starBSSN Baumgarte-Shapiro-Shibata-NakamuraCBM Compact binary mergersCMB Cosmic microwave backgroundDM Dark matterECO Exotic Compact ObjectEFT Effective Field theoryEMRI Extreme-mass-ratio inspiralEOB Effective One Body modelEOS Equation of stateeV electron VoltGR General RelativityGSF Gravitational self-forceGRB Gamma-ray burstGW Gravitational WaveHMNS Hypermassive neutron starIMBH Intermediate-mass black holeIVP Initial Value ProblemLVC LIGO Scientific and Virgo CollaborationsMBH Massive black holeNK Numerical kludge modelNSB Neutron star binaryNS Neutron starNR Numerical RelativityPBH Primordial black holePN Post-NewtonianPM Post-MinkowskianQNM Quasinormal modessBH Black hole of stellar originSGWB Stochastic GW backgroundSM Standard ModelSMBBH Supermassive binary black holeSOBBH Stellar-origin binary black holeSNR Signal-to-noise ratioST Scalar-tensor

CONTENTS 5

Contents

Chapter I: The astrophysics of compact object mergers:prospects and challenges 11

1 Introduction 11

2 LIGO and Virgo Observations of Binary Black Hole Mergers and aBinary Neutron Star 12

3 Black hole genesis and archaeology 143.1 Black Hole Genesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Black Hole Binaries: the difficulty of pairing . . . . . . . . . . . . . . . . 19

4 The formation of compact object mergers through classical binarystellar evolution 204.1 Stellar-origin black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1.1 Single star evolution . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.2 Binary star evolution . . . . . . . . . . . . . . . . . . . . . . . . . 234.1.3 Reconciling observations and theory . . . . . . . . . . . . . . . . . 24

4.2 BNS mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Mixed BH-NS mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Dynamical Formation of Stellar-mass Binary Black Holes 315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Merger rate estimates in dynamical channels . . . . . . . . . . . . . . . . 315.3 Advances in numerical methods in dynamical modeling . . . . . . . . . . 34

5.3.1 Direct N -body integration . . . . . . . . . . . . . . . . . . . . . . 345.3.2 Monte-Carlo methods . . . . . . . . . . . . . . . . . . . . . . . . . 345.3.3 Secular Symplectic N -body integration . . . . . . . . . . . . . . . 355.3.4 Semianalytical methods . . . . . . . . . . . . . . . . . . . . . . . 35

5.4 Astrophysical interpretation of dynamical sources . . . . . . . . . . . . . 365.4.1 Mass distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4.2 Spin distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4.3 Eccentricity distribution . . . . . . . . . . . . . . . . . . . . . . . 375.4.4 Sky location distribution . . . . . . . . . . . . . . . . . . . . . . . 375.4.5 Smoking gun signatures . . . . . . . . . . . . . . . . . . . . . . . 37

6 Primordial Black Holes and Dark Matter 386.1 Motivation and Formation Scenarios. . . . . . . . . . . . . . . . . . . . . 386.2 Astrophysical probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.3 Discriminating PBHs from astrophysical BHs . . . . . . . . . . . . . . . 42

CONTENTS 6

7 Formation of supermassive black hole binaries in galaxy mergers 42

8 Probing supermassive black hole binaries with pulsar timing arrays 468.1 The Gravitational-Wave Background . . . . . . . . . . . . . . . . . . . . 478.2 Continuous Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . 49

9 Numerical Simulations of Stellar-mass Compact Object Mergers 499.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499.2 Recent results for binary neutron star mergers . . . . . . . . . . . . . . . 51

9.2.1 Gravitational waves and remnant properties . . . . . . . . . . . . 519.2.2 Matter ejection and electromagnetic counterparts . . . . . . . . . 529.2.3 GW170817 and its counterparts . . . . . . . . . . . . . . . . . . . 53

9.3 Recent results for black hole-neutron star mergers . . . . . . . . . . . . . 549.4 Perspectives and future developments . . . . . . . . . . . . . . . . . . . . 54

10 Electromagnetic Follow-up of Gravitational Wave Mergers 5610.1 The High-energy Counterpart . . . . . . . . . . . . . . . . . . . . . . . . 5710.2 The Optical and Infrared Counterpart . . . . . . . . . . . . . . . . . . . 5710.3 The Radio Counterpart . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5910.4 Many Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

11 X-ray and gamma-ray binaries 60

12 Supermassive black hole binaries in the cores of AGN 6212.1 Modeling electromagnetic signatures of merging SMBBHs . . . . . . . . . 65

13 Cosmology and cosmography with gravitational waves 6613.1 Standard sirens as a probe of the late universe . . . . . . . . . . . . . . . 66

13.1.1 Redshift information . . . . . . . . . . . . . . . . . . . . . . . . . 6713.1.2 Standard sirens with current GW data: GW170817 . . . . . . . . 6913.1.3 Cosmological forecasts with standard sirens . . . . . . . . . . . . 6913.1.4 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

13.2 Interplay between GWs from binaries and from early-universe sources . . 7213.2.1 Status and future prospects . . . . . . . . . . . . . . . . . . . . . 73

Chapter II: Modelling black-hole sources of gravitationalwaves: prospects and challenges 75

1 Introduction 75

2 Perturbation Methods 762.1 Recent developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

CONTENTS 7

2.1.1 First-order gravitational self-force for generic orbits in Kerrspacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.1.2 Extraction of gauge-invariant information . . . . . . . . . . . . . 782.1.3 Synergy with PN approximations, EOB theory, and NR . . . . . . 792.1.4 New and efficient calculational approaches . . . . . . . . . . . . . 792.1.5 Cosmic censorship . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.2 Remaining challenges and prospects . . . . . . . . . . . . . . . . . . . . . 802.2.1 Efficient incorporation of self-force information into waveform

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.2.2 Producing accurate waveform models: self-consistent evolution

and second-order gravitational self-force . . . . . . . . . . . . . . 812.2.3 Gravitational Green Function . . . . . . . . . . . . . . . . . . . . 822.2.4 Internal-structure effects . . . . . . . . . . . . . . . . . . . . . . . 822.2.5 EMRIs in alternative theories of gravity . . . . . . . . . . . . . . 822.2.6 Open tools and datasets . . . . . . . . . . . . . . . . . . . . . . . 83

3 Post-Newtonian and Post-Minkowskian Methods 833.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.2 Recent developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2.1 4PN equations of motion for non-spinning compact-object binaries 843.2.2 Spin effects in the binary dynamics and gravitational waveform . 853.2.3 Comparisons to perturbative gravitational self-force calculations . 863.2.4 First law of compact binary mechanics . . . . . . . . . . . . . . . 87

3.3 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 Numerical Relativity and the Astrophysics of Black Hole Binaries 894.1 Current status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3 Numerical Relativity and GW observations . . . . . . . . . . . . . . . . . 93

5 Numerical relativity in fundamental physics 945.1 Particle laboratories in outer space . . . . . . . . . . . . . . . . . . . . . 955.2 Boson stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.3 Compact objects in modified theories of gravity . . . . . . . . . . . . . . 985.4 High-energy collisions of black holes . . . . . . . . . . . . . . . . . . . . . 1005.5 Fundamental properties of black holes and non-asymptotically flat

spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 Effective-One-Body and Phenomenological models 1026.1 Effective-one-body models . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.2 Phenomenological (Phenom) models . . . . . . . . . . . . . . . . . . . . . 1046.3 Remaining challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4 Tests of GR with EOB and Phenom models . . . . . . . . . . . . . . . . 107

CONTENTS 8

7 Source modelling and the data-analysis challenge 1077.1 Overview of current data analysis methods . . . . . . . . . . . . . . . . . 107

7.1.1 Unmodelled searches . . . . . . . . . . . . . . . . . . . . . . . . . 1077.1.2 Template based searches . . . . . . . . . . . . . . . . . . . . . . . 1087.1.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2 Challenges posed by future detectors . . . . . . . . . . . . . . . . . . . . 1097.2.1 Third-generation ground-based detectors . . . . . . . . . . . . . . 1097.2.2 The Laser Interferometer Space Antenna . . . . . . . . . . . . . . 110

7.3 Computationally efficient models . . . . . . . . . . . . . . . . . . . . . . 1127.3.1 Reduced-order models . . . . . . . . . . . . . . . . . . . . . . . . 1127.3.2 Kludges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8 The view from Mathematical GR 1148.1 Quasi-Kerr remnants? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158.2 Quasi-normal ringing? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158.3 Quasi-local masses, momenta and angular momenta? . . . . . . . . . . . 1168.4 Quasi-mathematical numerical relativity? . . . . . . . . . . . . . . . . . . 116

Chapter III: Gravitational waves and fundamental physics 118

1 Introduction 118

2 Beyond GR and the standard model 1182.1 Alternative theories as an interface between new physics and observations 1182.2 Scalar-tensor theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.3 Lorentz violations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1242.4 Massive gravity and tensor-tensor theories . . . . . . . . . . . . . . . . . 126

3 Detecting new fields in the strong gravity regime 1293.1 Gravitational wave propagation . . . . . . . . . . . . . . . . . . . . . . . 1303.2 Inspiral and parametrized templates . . . . . . . . . . . . . . . . . . . . . 131

3.2.1 Post-Newtonian parametrisation . . . . . . . . . . . . . . . . . . . 1313.2.2 Parameterized post-Einsteinian (ppE) formalism . . . . . . . . . . 1323.2.3 Phenomenological waveforms . . . . . . . . . . . . . . . . . . . . 132

3.3 Ringdown and black hole perturbations beyond General Relativity . . . . 1333.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343.3.2 Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.3.3 A parametrised ringdown approach? . . . . . . . . . . . . . . . . 136

3.4 Merger, numerics, and complete waveforms beyond General Relativity . . 1373.4.1 Initial value formulation and predictivity beyond GR . . . . . . . 1373.4.2 Well-posedness and effective field theories . . . . . . . . . . . . . 1383.4.3 Numerical simulations in alternative theories . . . . . . . . . . . . 139


4 The nature of Compact Objects 1404.1 No-hair theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.2 Non-Kerr black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

4.2.1 Circumventing no-hair theorems . . . . . . . . . . . . . . . . . . . 1424.2.2 Black holes with scalar hair . . . . . . . . . . . . . . . . . . . . . 1444.2.3 Black hole scalarization . . . . . . . . . . . . . . . . . . . . . . . . 1444.2.4 Black holes in theories with vector fields and massive/bimetric

theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444.2.5 Black holes in Lorentz-violating theories . . . . . . . . . . . . . . 145

4.3 Horizonless exotic compact objects . . . . . . . . . . . . . . . . . . . . . 1464.4 Testing the nature of compact objects . . . . . . . . . . . . . . . . . . . . 149

4.4.1 EM diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494.4.2 GW diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

5 The dark matter connection 1545.1 BHs as DM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2 The DM Zoo and GWs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.3 The cosmological perspective on DM: scalar field dark matter . . . . . . 157

5.3.1 Cosmological background dynamics . . . . . . . . . . . . . . . . . 1585.3.2 Cosmological linear perturbations . . . . . . . . . . . . . . . . . . 1585.3.3 Cosmological non-linear perturbations . . . . . . . . . . . . . . . 158

5.4 The DM environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1595.5 Accretion and gravitational drag . . . . . . . . . . . . . . . . . . . . . . . 1615.6 The merger of two dark stars . . . . . . . . . . . . . . . . . . . . . . . . 1625.7 Non-perturbative effects: superradiance and spin-down. . . . . . . . . . . 1625.8 Pulsar timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.9 Mini-galaxies around BHs . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.10 GW detectors as DM probes . . . . . . . . . . . . . . . . . . . . . . . . . 164

Postscript 165


Preface

The long-held promise of gravitational-wave astronomy as a new window ontothe universe has finally materialized with the dramatic discoveries of the LIGO-Virgocollaboration in the past few years. We have taken but the first steps along a new,exciting avenue of exploration that has now opened before us. The questions we willtackle in the process are cross-cutting and multidisciplinary, and the answers we will getwill no doubt reshape our understanding of black-hole-powered phenomena, of structureformation in the universe, and of gravity itself, at all scales.

The harvesting of useful information from gravitational-wave (GW) signals andthe understanding of its broader implications demand a cross-disciplinary effort. Whatexactly will GWs tell us about how, when and in which environment black holes wereformed? How fast do black holes spin and how have some of them grown to becomesupermassive? GWs from merging black holes probe the environment in which theyreside, potentially revealing the effect of dark matter or new fundamental degrees offreedom. The analysis of GWs will allow for precise tests of General Relativity, andof the black hole paradigm itself. However, to be able to collect and interpret theinformation encoded in the GWs, one has to be equipped with faithful and accuratetheoretical models of the predicted waveforms. To accomplish the far-reaching goalsof gravitational-wave science it is of paramount importance to bring together expertiseover a very broad range of topics, from astrophysics and cosmology, through general-relativistic source modelling to particle physics and other areas of fundamental science.

In 2016, a short time before the announcement of the first gravitational-wavedetection, a cross-disciplinary initiative in Europe led to the establishment of thenew COST networking Action on “Black holes, gravitational waves and fundamentalphysics” (“GWverse”). GWverse aims to maintain and consolidate leadership in black-hole physics and gravitational-wave science, linking three scientific communities that arecurrently largely disjoint: one specializing in gravitational-wave detection and analysis,another in black-hole modelling (in both astrophysical and general-relativistic contexts),and a third in strong-gravity tests of fundamental physics. The idea is to form a single,interdisciplinary exchange network, facilitating a common language and a frameworkfor discussion, interaction and learning. The Action will support the training of thenext generation of leaders in the field, and the very first “native” GW/multi-messengerastronomers, ready to tackle the challenges of high-precision GW astronomy with groundand space-based detectors.

Leor BarackVitor CardosoSamaya NissankeThomas Sotiriou


Chapter I: The astrophysics of compact object mergers:prospects and challenges

Editor: Samaya Nissanke

1. Introduction

In the last two years, strong-field gravity astrophysics research has been undergoinga momentous transformation thanks to the recent discoveries of five binary black hole(BBH) mergers that were observed in gravitational waves (GWs) by the LIGO and Virgodetectors. This was compounded last year by the multi-messenger discovery of a binaryneutron star (BNS) merger measured in both GWs and detected in every part of theelectromagnetic (EM) spectrum, allowing us to place compact object mergers in theirfull astrophysical context. These measurements have opened up an entirely new windowonto the Universe, and given rise to a new rapidly growing and observationally-drivenfield of GW astrophysics.

Despite the multiple scientific breakthroughs and “firsts” that these discoveriessignify, the measured properties of the BBH and BNS mergers have immediatelybought up accompanying challenges and pertinent questions to the wider astrophysicscommunity as a whole. Here, we aim to provide an up-to-date and encompassing reviewof the astrophysics of compact object mergers and future prospects and challenges.Section 2 first introduces and briefly details the LIGO and Virgo observations of BBHand BNS mergers. Section 3 then discusses the astrophysics of BHs, in particular BBHs,from their genesis to archeology, for BHs that span more than ten decades in their massrange. In the case of stellar-mass BBH mergers, Section 4 details the formation ofcompact binary mergers, in particular BBHs, through isolated stellar binary evolution.Section 5 then reviews how one could dynamically form such events in order to explainthe observed merger rate and distribution of masses, mass ratios, and spins. Section 6explores the intriguing possibility that at least a fraction of dark matter (DM) in theUniverse is in the form of primordial BHs (PBHs), an area that has recently beeninvigorated by the recent LIGO and Virgo observations of BBH mergers. Section 7presents an overview on the formation of supermassive BBHs through galaxy mergers,and Section 8 introduces efforts underway to probe the astrophysics of such supermassiveBBHs with pulsar timing arrays. Turning our attention to the mergers themselves asmulti-messenger sources, Section 9 reviews the state-of-the-art numerical modelling ofcompact object mergers, in particular, systems with NSs in which we have alreadyobserved accompanying EM radiation. For detailed and exhaustive discussion of themodelling of BBH mergers, we refer the reader to Chapter II. Section 10 provides asummary of the observational efforts by a wide range of facilities and instruments infollowing up GW mergers in light of the first BNS merger discovery measured in bothGWs and EM. Focusing entirely on EM observations, Section 12 reviews observations ofactive galactic nuclei as probes of BBH systems and Section 11 concludes by summarising


recent advances in high-energy observations of X-ray binaries. Finally, Section 13provides an extensive review on how observations of GWs can impact the field ofcosmology, that is, in our understanding of the origins, evolution and fate of the Universe.

2. LIGO and Virgo Observations of Binary Black Hole Mergers and aBinary Neutron Star

Contributors: E. Porter, M. Hendry, I. S. Heng

On September 15th 2014 the discovery of GWs from the merger of two BHs duringthe first advanced detectors era run, commonly called O1, by the two LIGO observatoriesheralded the dawn of GW astronomy [1]. This event was quickly followed up by twoother BBH mergers: one of lower significance on October 12th, 2015, and another onDecember 26th, 2015 [2, 3]; see Table 1 for the source properties of the published GWmergers. These detections, as exemplified by this white paper, have had a major impacton the fields of astrophysics and fundamental physics [3–8].

The detection of GWs from only BBH mergers from all O1 detections has hadsignificant ramifications on our understanding of astrophysical populations [3, 6, 9, 10].The detected BHs were more massive than any BHs that had been previously detectedin low mass X-ray binaries, requiring a re-evaluation of the models of stellar evolutionin binary systems [5]. From just these three events, the LIGO Scientific and Virgocollaborations (LVC) constrained the rate of BBH mergers to between 9-240 Gpc−3

yr−1 [3] (see [11] for an updated BBH merger rate of 12-213 Gpc−3 yr−1). The non-detection of BNSs and NS-BH binaries allowed constraints of < 12, 600 Gpc−3 yr−1 and< 3, 600 Gpc−3 yr−1 respectively [6]. At the time of this run, the LVC had over 60 MOUssigned with external telescopes, satellites and neutrino detectors. No EM counterpartswere found relating to the BBH mergers [12–14].

To detect and extract astrophysical information, GW astronomy uses the method ofmatched filtering [15]. This method is the optimal linear filter for signals buried in noise,and is very much dependent on the phase modelling of a GW template. Within the LVC,the GW templates are constructed using both analytical and numerical relativity [16].In this case, the phase evolution of the template is a function of a number of frequencydependent coefficients. Alternative theories of gravity predict that these coefficientsshould be individually modified if general relativity (GR) is not the correct theory ofgravity. While GR predicts specific values for these coefficients, one can treat eachcoefficient as a free variable and use Bayesian inference to test for deviations in thevalues of the parameters from the nominal GR value. All tests conducted by the LVCdisplayed no deviations from GR [3, 11, 17, 18]

In addition, searches for generic GW transients, or GW-bursts, typically do notrequire a well-known or accurate waveform model and are robust against uncertaintiesin the GW signature. GW-burst searches are designed to detect transients withdurations between 10−3 − 10 seconds with minimal assumptions about the expectedsignal waveform. Such searches are, therefore, sensitive to GW transients from a wide


range of progenitors, ranging from known sources such as BBH mergers to poorly-modeled signals such as core-collapse supernovae as well as transients that have yetto be discvered. An overview of GW-burst searches performed by LVC can be foundhere [19]. Both GW-burst and compact binary coalescences (CBC) searches detectedthe first GW signal from BBH mergers, GW150914.

In November 2016, the second Advanced Era Observation run, O2, began. Onceagain, in January and June 2017, two BBH mergers were observed by the two LIGOdetectors [11, 18]. At the end of July 2017, the Advanced Virgo detector joined theglobal network of detectors. On August 14th, all three detectors observed the merger ofa BBH system. In previous detections, using only the two LIGO detectors, the sourceswere located to 1000s of square degrees in the sky. In this case, due to the additionof Advanced Virgo, this system was localised to within 60 square degrees. While notgreatly advancing our understanding of the formation mechanisms of such systems,this detection did have a major effect in the field of fundamental physics. Due to themisalignment of the three detectors, for the first time we were able to test the tensorialnature of GWs. This event allowed the LVC to conclude that the GW signals weretensorial in nature, as is predicted by GR [18].

Burst searches were also used as an independent analysis to complement matchedfiltering analyses for the detection of GW170104 [11]. Burst searches further identifieda coherent signal, corresponding to GW170608, with a false-alarm rate of 1 in ∼ 30years [20] and validated the detection of GW170814 with a false-alarm rate < 1 in 5900years [18]. Note that, given the “unmodelled” nature of burst searches, the estimatedevent significances from burst searches tend to be lower than matched-filtered searchesfor the same event, especially for lower-mass compact binary signals.

On August 17th, the first BNS merger was observed by the LIGO and Virgodetectors [21]. This event was very quickly associated with a short gamma-ray burst(sGRB) detected by both the Fermi and Integral satellites [22]. Within 10 hours, thehost galaxy had been optically identified. Within 16 days, the source had been identifiedacross all bands of the EM spectrum. This single event heralded the true beginning ofmulti-messenger astronomy, and raised as many questions as it answered.

While confirming the hypothetical link between BNS mergers and sGRBs, the delaybetween the gamma and X-ray signals (9 days) suggested that not all sGRBs are thesame [23]. This fact generated a number of studies regarding equation of state models,and the possible remnant of such mergers. This one event also allowed the LVC toupdate the BNS event rate from < 12, 600 Gpc−3 yr−1 in O1, to 320-4740 Gpc−3 yr−1

in O2 [21].Perhaps, the most interesting results from this event concern fundamental physics.

The delay between the detection of GWs and gamma-rays was 1.74 seconds. Thisplaces a bound on the difference between the speed of light and the speed of GWs of3 × 10−15 ≤ |∆c/c| ≤ 7 × 10−16 [23]. This single result has implications for certainalternative theories of gravity. For instance, the fact that GWs seem to travel at thesame speed as that of strongly constrains the family of alternative theories of gravity


GW150914 GW151226 LVT151012 GW170104 GW170608 GW170814 GW170817m1/ M 36.2+5.2

−3.8 14.2+8.3−3.7 23+18

−6 31.2+8.4−6.0 12+7

−2 30.5+5.7−3.0 (1.36, 1.60)

m2/ M 29.1+3.7−4.4 7.5+2.3

−2.3 13+4−5 19.4+5.3

−5.9 7+2−2 25.3+2.8

−4.2 (1.16, 1.36)M/ M 28.1+1.8

−1.5 8.88+0.33−0.28 15.1+1.4

−1.1 21.1+2.4−2.7 7.9+0.2

−0.2 24.1+1.4−1.1 1.186+0.001

−0.001q 0.81+0.17

−0.20 0.52+0.40−0.29 0.57+0.38

−0.37 0.62+− 0.6+0.3

−0.4 0.83+− (0.73, 1)

Mf/ M 62.3+3.7−3.1 20.8+6.1

−1.7 35+14−4 48.7+5.7

−4.6 18.0+4.8−0.9 53.2+3.2

−2.5 −−−χeff −0.06+0.14

−0.29 0.21+0.20−0.10 0.03+0.31

−0.20 −0.12+0.21−0.30 0.07+0.23

−0.09 0.06+0.12−0.12 0.00+0.02

−0.01af 0.68+0.05

−0.06 0.74+0.06−0.06 0.66+0.09

−0.10 0.64+0.09−0.20 0.69+0.04

−0.05 0.70+0.07−0.05 −−−

DL / Mpc 420+150−180 440+180

−190 1020+500−490 880+450

−390 340+140−140 540+130

−210 40z 0.090+0.029

−0.036 0.094+0.035−0.039 0.201+0.086

−0.091 0.18+0.08−0.07 0.07+0.03

−0.03 0.11+0.03−0.04 0.0099

Table 1. Source properties of the published BBH and BNS discoveries (June 2018)by the LIGO and Virgo detectors

that require v+, v× 6= vlight (e.g., beyond Horndeski, quartic/quintic Galileon, Gauss-Bonnet, if they are supposed to explain cosmology), as well as theories that predicta massive graviton. Furthermore, by investigating the Shapiro delay, the GW170817detection also rules out MOND and DM emulator MOND-like theories (e.g., TeVeS), asaccording to these theories, the GWs would have arrived 1000 days after the gamma-raydetection.

The detection of GWs by the Advanced LIGO and Advanced Virgo detectors havehad a major effect on our understanding of the Universe, sparking the fields of GW andmulti-messenger astronomy and cosmology [22, 24]. It is becoming increasingly clearthat combining EM and GW information will be the only way to better explain observedphenomena in our Universe. The third Advanced Detector Observation run (O3) willbegin in early 2019, and will run for a year [14]. We expect the detected events to bedominated by BBH mergers at a rate of one per week. However, we also expect onthe order of ten BNS events during this time, and possibly a NS-BH discovery (andpotentially more than one such system). Given the effects of one GW detection on bothastrophysics and fundamental physics, we expect O3 to fundamentally change our viewof the Universe.

3. Black hole genesis and archaeologyContributors: M. Colpi and M. Volonteri

3.1. Black Hole Genesis

Gravity around BHs is so extreme that gravitational energy is converted into EMand kinetic energy with high efficiency, when gas and/or stars skim the event horizonof astrophysical BHs. Black holes of stellar origin (sBHs) with masses close to thoseof known stars power galactic X-ray sources in binaries, while supermassive black holes(SMBHs) with masses up to billions of solar masses power luminous quasars and activenuclei at the centre of galaxies. BHs are key sources in EM in our cosmic landscape.

According to General Relativity (GR), Kerr BHs, described by their massMBH and


spin vector S = χspin GMBH/c (with−1 ≤ χspin ≤ 1) are the unique endstate of unhaltedgravitational collapse. Thus understanding astrophysical BHs implies understanding theconditions under which gravitational equilibria lose their stability irreversibly. The chiefand only example we know is the case of NSs which can exist up to a maximum massMNS

max, around 2.2 M − 2.6 M. No baryonic microphysical state emerges in nuclearmatter, described by the standard model, capable to reverse the collapse to a BH state,during the contraction of the iron core of a supernova progenitor. The existence ofMNS

maxis due to the non linearity of gravity which is sourced not only by the mass “charge”but also by pressure/energy density, according to the Oppenheimer-Volkoff equation.Thus, sBHs carry a mass exceedingMNS

max. Discovering sBHs lighter than this value (notknown yet to high precision) would provide direct evidence of the existence of PBHsarising from phase transitions in the early universe.

As of today, we know formation scenarios in the mass range between 5 M−40 M,resulting from the core-collapse of very massive stars. The high masses of the sBHsrevealed by the LVC, up to 36 M, hint formation sites of low-metallicity, Z∗, below0.5% of the solar value Z = 0.02 [25–27]. Theory extends this range up to about40−60 M [28] and predicts the existence of a gap, between about 60 <∼MBH/ M <∼ 150,since in this window pair instabilities during oxygen burning lead either to substantialmass losses or (in higher mass stellar progenitors) the complete disruption of the star [29–31]. sBHs heavier than 150 M can form at Z < 1%Z, if the initial mass function ofstars extends further out, up to hundreds of solar masses.

The majestic discovery of BBHs, detected by LVC interferometers [1, 2, 5, 11, 18,20], at the time of their coalescence further indicates, from an astrophysical standpoint,that in nature sBHs have the capability of pairing to form binary systems, contractedto such an extent that GW emission drives their slow inspiral and final merger, onobservable cosmic timescales. As GWs carry exquisite information on the individualmasses and spins of the BHs, and on the luminosity distance of the source, detectinga population of coalescing sBHs with LVC in their advanced configurations, and withthe next-generation of ground-based detectors [32, 33], will let us reconstruct the massspectrum and evolution of sBHs out to very large redshifts.

Observations teach us that astrophysical BHs interact with their environment, andthat there are two ways to increase the mass: either through accretion, or through amerger, or both. These are the two fundamental processes that drive BH mass and spinevolution. Accreting gas or stars onto BHs carry angular momentum, either positiveor negative, depending on the orientation of the disk angular momentum relative tothe BH spin. As a consequence the spin changes in magnitude and direction [34–36].In a merger, the spin of the new BH is the sum of the individual and orbital angularmomenta of the two BHs, prior to merging [37, 38]. An outstanding and unansweredquestion is can sequences of multiple accretion-coalescence events let sBHs grow, in some(rare) cases, up to the realm of SMBHs? If this were true, the “only” collapse to a BH

∗ In astrophysics “metallicity” refers to the global content of heavy elements above those produced byprimordial nucleosynthesis


occurring in nature would be driven by the concept of instability of NSs at MNSmax.

SMBHs are observed as luminous quasars and active galactic nuclei, fed by accretionof gas [39], or as massive dark objects at the centre of quiescent galaxies whichperturb the stellar and/or gas dynamics in the nuclear regions [40]. The SMBH massspectrum currently observed extends from about 5× 104 M (the SMBH in the galaxyRGG118 [41]) up to about 1.2 × 1010 M (SDSS J0100+2802 [42]), as illustrated inFigure 1. The bulk of active and quiescent SMBHs are nested at the centre of theirhost galaxies, where the potential well is the deepest. The correlation between theSMBH mass M• and the stellar velocity dispersion σ in nearby spheroids, and even indisk/dwarf galaxies [43] hints towards a concordant evolution which establishes in thecentre-most region controlled by powerful AGN outflows. Extrapolated to lower massdisk or dwarf galaxies, this correlation predicts BH masses of M• ∼ 103 M at σ as lowas 10 km s−1, typical of nuclear star clusters (globular clusters) [44]. We remark thatonly BHs of mass in excess of 103 M can grow a stellar cusp. The lighter BHs wouldrandom walk, and thus would have a gravitational sphere of influence smaller than themean stellar separation and of the random walk mean pathlength.

Observations suggest that SMBHs have grown in mass through repeated episodesof gas accretion and (to a minor extent) through mergers with other BHs. This complexprocess initiates with the formation of a seed BH of yet unknown origin [45]. Theconcept of seed has emerged to explain the appearance of a large number of SMBHs ofbillion suns at z ∼ 6, shining when the universe was only 1 Gyr old [46]. Furthermore,the comparison between the local SMBH mass density, as inferred from the M• − σ

relation, with limits imposed by the cosmic X-ray background light, resulting fromunresolved AGN powered by SMBHs in the mass interval between 108−9 M, indicatesthat radiatively efficient accretion played a large part in the building of SMBHs belowz ∼ 3, and that information is lost upon their initial mass spectrum [47]. Thus, SMBHsare believed to emerge from a population of seeds of yet unconstrained initial mass, in amass range intermediate between those of sBHs and SMBHs, about 102 M to 105 M,and therefore they are sometimes dubbed Intermediate-mass BHs (IMBHs).

Seeds are IMBHs that form “early” in cosmic history (at redshift z ∼ 20, when theuniverse was only 180 Myr old). They form in extreme environments, and grow overcosmic time by accretion and mergers. Different formation channels have been proposedfor the seeds [45, 48, 49]. Light seeds refer to IMBHs of about 100 M that form from therelativistic collapse of massive Pop III stars, but the concept extends to higher masses,up to ∼ 103 M. These seeds likely arise from runaway collisions of massive stars indense star clusters of low metallicity [50, 51], or from mergers of sBHs in star clusterssubjected to gas-driven evolution [52]. The progenitors of light seeds are massive stars.However, they could be also the end result of repeated mergers among BHs [53]. Findingmerging sBHs with the LVC detectors with masses in the pair instability gap would be aclear hint of a second generation of mergers resulting from close dynamical interactions.

Accretion on sBHs occurs in X-ray binaries, and there is no evidence of accretionfrom the interstellar medium onto isolated sBHs in the Milky Way. But, in gas-rich,


pair i

nsta

bility

gap

SEEDS

log

(mas

s dist

ribut

ion)

sBHs

SMBHs

RGG118 SgR A* J0100+2802

GW150914

log (M/M⊙)

Figure 1. Cartoon illustrating the BH mass spectrum encompassing the wholeastrophysical relevant range, from sBHs to SMBHs, through the unexplored (light-green) zone where BH seeds are expected to form and grow. Vertical black-linesdenote the two sBH masses in GW150914, the mass M• of RGG118 (the lightestSMBH known as of today in the dwarf galaxy RG118), of SgrA* in the Milky Way,and of J0100+2802 (the heaviest SMBH ever recorded). The mass distribution ofsBHs, drawn from the observations of the Galactic sBH candidates, has been extendedto account for the high-mass tail following the discovery of GW150914. The minimum(maximum) sBHs is set equal to 3 M (60 M), and the theoretically predicted pair-instability gap is depicted as a narrow darker-grey strip. The SMBH distribution hasbeen drawn scaling their mass according to the local galaxy mass function and M•-σcorrelation. The decline below ∼ 105 M is set arbitrarily: BH of ∼ 104−5 M maynot be ubiquitous in low-mass galaxies as often a nuclear star cluster is in place in thesegalaxies, which may or may not host a central IMBH [54]. The black stars and dashedtracks illustrate the possibility that a SMBH at high redshift forms as sBH-only (bornon the left side of the sBH gap) or as light seed (on the right of the gap) which thengrows through phases of super-Eddington accretion [55]. The red circle and dottedtrack illustrates the possibility of a genetic divide between sBHs and SMBHs, and thata heavy seed forms through the direct collapse of a supermassive protostar in a metalfree, atomic-hydrogen cooling, DM halo [48, 56]. The seed later grows via gas accretionand mergers with SMBHs in other black halos.


dense environments characteristic of galaxy halos at high redshifts, single sBHs mightaccrete to grow sizably, despite their initial small gravitational sphere of influence, ifspecific dynamical conditions are met. For instance, in rare cases they may be capturedin dense gas clouds within the galaxy [55]. Another possibility is that a sBH forms atthe very center of the galaxy, where large inflows may temporarily deepen the potentialwell and allow it to grow significantly. This “winning sBH” must be significantly moremassive than all other sBHs in the vicinity to avoid being ejected by scatterings and tobe retained at the center of the potential well by dynamical friction. Similar conditionscan also be present in nuclear star clusters characterized by high escape velocities. Afterejection of the bulk of the sBHs, the only (few) remaining isolated BH can grow by tidallydisrupting stars and by gas accretion [57] sparking their growth to become an IMBH.

Heavy seeds refer instead to IMBHs of about 104−5 M resulting from the monolithiccollapse of massive gas clouds, forming in metal-free halos with virial temperaturesTvir >∼ 104 K, which happen to be exposed to an intense H2 photodissociating ultravioletflux [49, 56, 58–60]. These gas clouds do not fragment and condense in a single massiveproto-star which is constantly fueled by an influx of gas that lets the proto-star growlarge and massive. Then, the star contracts sizably and may form a quasi-star [61],or it may encounter the GR instability that leads the whole star to collapse directlyinto a BH. Heavy seeds might also form in major gas-rich galaxy mergers over a widerrange of redshifts, as mergers trigger massive nuclear inflows [62]. Figure 1 is a cartoonsummarising the current knowledge of BHs in our Universe, and the link that may existbetween sBHs and SMBHs, which is established by seed BHs along the course of cosmicevolution.

The seeds of the first SMBHs are still elusive to most instruments that exist today,preventing us to set constraints on their nature. Seed BHs are necessarily a transientpopulation of objects and inferring their initial mass function and spin distributionfrom observations is possible only if they can be detected either through EM or GWobservations at very high z, as high as ∼ 20 (even z ∼ 40 as discussed recently). Since,according to GR, BHs of any flavour captured in binaries are loud sources of GWs atthe time of their merging, unveiling seeds and MBHs through cosmic ages via their GWemission at coalescence would provide unique and invaluable information on the BHgenesis and evolution. The Gravitational Wave Universe is the universe we can senseusing GWs as messengers [63, 64]. In this universe, BBHs are key sources carryinginvaluable information on their masses, spins and luminosity distance that are encodedin the GW signal. There is one key condition that needs to be fulfilled: that the BHswe aim at detecting pair and form a binary with GW coalescence time smaller that theHubble time, possibly close to the redshift of their formation. This condition, enablingthe detection of seeds at very high redshifts, is extremely challenging to be fulfilled. BHsin binaries form at “large” separation. Thus, nature has to provide additional dissipativeprocesses leading to the contraction of the BBH down to the scale where GWs drive theinspiral. This requires a strong coupling of the two BHs with the environment, beforeand after forming a binary system. As we now discuss, understanding this coupling


is a current challenge in contemporary astrophysics, cosmology and computationalphysics [65].

3.2. Black Hole Binaries: the difficulty of pairing

Due to the weakness of gravity, BBH inspirals driven by GW emission occur on atimescale:

tcoal = 5c5

256G3G(e)(1− e2)7/2 a4

ν M3BBH

= 5 · 24

256 G(e)(1− e2)7/2GMBBH

νc3 a4, (1)

where MBBH is the total mass of the BBH, a and e the semi-major axis and eccentricityrespectively (G(e) a weak function of e, and G(0) = 1) and ν = µ/MBBH the symmetricmass ratio (ν = 1/4 for equal mass binaries), with µ the reduced mass of the binary.The values of a and e at the time of formation of the binary determine tcoal, and this isthe longest timescale. A (circular) binary hosting two equal-mass seed BHs of 103 M(MBBH = 105 M) would reach coalescence in 0.27 Gyrs, corresponding to the cosmictime at redshift z ∼ 15, if the two BHs are at an initial separation of a ∼ ν1/44.84×104RG

( ν1/4 1.5× 104RG) corresponding to a ∼ ν1/4 0.1 AU, (ν1/4 30 AU). For the case of twoequal-mass MBHs of 106 M coalescing at z ∼ 3 (close to the peak of the star-formationrate and AGN rate of activity in the universe) a ∼ ν1/44.84 × 106RG correspondingto about one milli-parsec. These are tiny scales, and to reach these separations thebinary needs to harden under a variety of dissipative processes. The quest for efficientmechanisms of binary shrinking, on AU-scales for sBHs and BH seeds, and sub-galacticscales for MBHs, make merger rate predictions extremely challenging, as Nature hasto set rather fine-tuned conditions for a BBH to reach these critical separations. Onlybelow these critical distances the binary contracts driven by GW emission. The mergeroccurs when the GW frequency (to leading order equal to twice the Keplerian frequency)reaches a maximum value,

fmaxGW ∼

c3

π63/2GMBBH= 4.4× 103

(M

MBBH

)Hz. (2)

This frequency fmaxGW scales with the inverse of the total mass of the binary, MBBH as it

is determined by the size of the horizon of the two BHs, at the time of coalescence.Coalescing sBHs in binaries occur in galactic fields [25, 66], or/and in stellar systems

such as globular clusters or/and nuclear star clusters [67–71]. Thus, sBHs describephenomena inside galaxies. Since there is a time delay between formation of the binaryand its coalescence, dictated by the efficiency of the hardening processes, sBHs canmerge in galaxies of all types, as in this lapse time that can be of the order of Gyrs, hostgalaxies undergo strong evolution. Instead, coalescing IMBHs, seeds of the SMBHs,formed in DM halos at high redshifts, and thus track a different environment. Formingin pristine gas clouds their pairing either requires in-situ formation, e.g., from fissioning


of rotating super-massive stars [72], or via halo-halo mergers on cosmological scalessubjected to rapid evolution and embedded in a cosmic web that feed baryons throughgaseous stream [73]. Coalescing MBHs refer exclusively to galaxy-galaxy mergers ofdifferent morphological types [65] occurring during the clustering of cosmic structures,which encompass a wide range of redshifts, from z ∼ 9 to z ∼ 0 passing through the eraof cosmic reionization, and of cosmic high noon when the averaged star formation ratehas its peak.

In the following Sections we describe in detail the different channels proposed forthe formation and pairing of BBHs at all scales. For each physical scenario we reviewthe state of the art, challenges and unanswered questions and the most promising linesof research for the future. Ample space is devoted to stellar mass objects (BNS andBH-NS binaries and BBHs, with a particular focus on the latter), for which we discussseparately the three main formation channels: pairing of isolated binaries in the field,the various flavors of dynamical formation processes, relics from the early universe.We then move onto discuss the state of the art of our understanding of MBH binarypairing and evolution, the current theoretical and observational challenges, and the roleof future surveys and pulsar timing arrays (PTAs) in unveiling the cosmic populationof these elusive systems.

4. The formation of compact object mergers through classical binary stellarevolution

Contributors: K. Belczynski, T. Bulik, T. M. Tauris, G. Nelemans

4.1. Stellar-origin black holes

The LIGO/Virgo detections of BBH mergers can be explained with stellar-originBHs [27] or by primordial BHs that have formed from density fluctuations right afterBig Bang [74], Stars of different ages and chemical compositions can form BHs andsubsequently BBH mergers. In particular, the first metal-free (population III) starscould have produced BBH mergers in the early Universe (z ≈ 10), while the local (z ≈0 − 2) Universe is most likely dominated by mergers formed by subsequent generationof more metal-rich population II and I stars [75]. The majority of population I/II stars(hereafter: stars) are found in galactic fields (∼ 99%) where they do not experiencefrequent or strong dynamical interactions with other stars. In contrast, some smallfraction of stars (∼ 1%) are found in dense structures like globular or nuclear clusters,in which stellar densities are high enough that stars interact dynamically with otherstars. Here, we briefly summarize basic concepts of isolated (galactic fields) stellar andbinary evolution that leads to the formation of BBH mergers.

4.1.1. Single star evolution Detailed evolutionary calculations with numerical stellarcodes that include rotation (like BEC, MESA or the Geneva code, e.g., [76–79]) allow


us to calculate the evolution of massive stars. Note that these are not (detailed)hydrodynamic nor multi-dimensional calculations (as such computations are wellbeyond current computing capabilities), but they solve the basic equations of stellarstructure/radiation, energy transport and element diffusion with corrections for effectsof rotation. These calculations are burdened with uncertainties in treatment of variousphysical processes (nuclear reaction rates, convection and mixing, transport of angularmomentum within a star, wind mass loss, pulsations and eruptions), yet progress is beingmade to improve on stellar modeling. Stellar models are used to predict the structureand physical properties of massive stars at the time of core-collapse, after nuclear fusionstops generating energy and (radiation) pressure that supports the star. This is also apoint in which transition (at latest) to hydrodynamical calculations is being made toassess the fate of a collapsing star [80–82].

For a star to form a BH, it is required that either the explosion engine is weak ordelayed (so energy can leak from the center of the collapsing star) or that the infallingstellar layers are dense and massive enough to choke the explosion engine adopted in agiven hydrodynamical simulation. In consequence, BHs form either in weak supernovaeexplosions (with some material that is initially ejected falling back and accreting ontothe BH or without a supernova explosion at all (in a so-called direct collapse). Notethat signatures of BH formation may already have been detected. For example, inSN 1987A there is no sign of a pulsar [83], although the pulsar may still appear whendust obscuration decreases or it simply beams in another direction. Further evidence isthe disappearance with no sign of a supernova of a 25 M supergiant star [84], althoughthis can be a potentially long period pulsating Mira variable star that will re-emergeafter a deep decline in luminosity.

Stellar evolution and core-collapse simulations favor the formation of BHs withmasses MBH ∼ 5 − 50 M and possibly with very high masses MBH & 135 M. Thelow-mass limit is set by the so-called “first mass gap”, coined after the scarcity ofcompact objects in mass range 2 − 5 M [85, 86]. However, this mass gap may benarrower than previously thought as potential objects that fill the gap are discovered[87–89]. The second gap arises from the occurrence of pair-instability SNe (PISN) asdiscussed below.

The first mass gap may be explained either by an observational bias indetermination of BH masses in Galactic X-ray binaries [90] or in terms of a timescale ofdevelopment of the supernova explosion engine: for short timescales (∼100 ms) a massgap is expected, while for longer timescales (∼1 s) a mass gap does not appear and NSsand BHs should be present in the 2−5 M mass range [91]. The mass threshold betweenNSs and BHs is not yet established, but realistic equations-of-state indicate that thisthreshold lies somewhere in range 2.0 − 2.6 M. The second limit at MBH ∼ 50 M iscaused by (pulsational) PISNe [92, 93]. Massive stars with He-cores in the mass range45 .MHe . 65 M are subject to pulsational PISNe before they undergo a core-collapse.These pulsations are predicted to remove the outer layers of a massive star (above theinner 40 − 50 M) and therefore this process limits the BH mass to ∼ 50 M. BHs


within these two limits (MBH ∼ 5− 50 M) are the result of the evolution of stars withan initial mass MZAMS ≈ 20−150 M. For high-metallicity stars (typical of stars in theMilky Way disk, Z = Z = 0.01−0.02; [94, 95]) BHs form up to ∼ 15 M, for medium-metallicity stars (Z = 10% Z) BHs form up to ∼ 30 M, while for low-metallicity stars(Z = 1% Z) BHs form up to ∼ 50 M [96, 97].

The remaining question is whether stars can form BHs above ∼ 50 M. Starswith He-cores in mass range: 65 . MHe . 135 M are subject to PISNe [92, 98, 99]that totally disrupts the star and therefore does not produce a BH. However, it isexpected that stars with He cores as massive as MHe & 135 M, although subject topair instability, are too massive to be disrupted and they could possibly form massiveBHs (MBH & 135 M). If these massive BHs exist, then second mass gap will emergewith no BHs in the mass range MBH ' 50 − 135 M [100–103]. If these massive BHsexist, and if they find their way into merging BBH binaries then GW observatorieswill eventually find them [103, 104]. The existence of very massive BHs will constrainthe extend of the stellar initial mass function (IMF) and wind mass-loss rates for themost massive stars (MZAMS > 300 M) that can produce these BHs. So far, there areno physical limitations for the existence of such massive stars [105, 106]. Note that themost massive stars known today are found in the LMC with current masses of ∼ 200 M[107].

BH formation may be accompanied by a natal kick. Natal kicks are observed forGalactic radio pulsars, that move significantly faster (with average 3-dimensional speedsof ∼ 400 km s−1, e.g., [108]) than their typical progenitor star (10− 20 km s−1). Thesehigh velocities are argued to be associated with some supernova asymmetry: eitherasymmetric mass ejection [109–111] or asymmetric neutrino emission [112–114]. Notethat neutrino kick models all require (possibly unrealistic) strong magnetic fields, andsimulations of core collapse without magnetic fields are unable to produce significantneutrino kicks. Naturally, in these simulations the authors find the need for asymmetricmass ejection to explain natal kicks (e.g., [110]). Although BH natal kicks as high asobserved for NSs cannot yet be observationally excluded, it is unlikely for BHs to receivesuch large natal kicks [115, 116]. It appears that some of the BHs may form withouta natal kick [117, 118], while some may form with a kick of the order of ∼ 100 km s−1

[116].The BH natal spin may simply depend on the angular momentum content of the

progenitor star at the time of core collapse. Massive stars are known to rotate; withthe majority of massive stars spinning at moderate surface velocities (about 90% at∼ 100 km s−1) and with some stars spinning rather rapidly (10% at ∼ 400 km s−1).During its evolution, a star may transport angular momentum from its interior toits atmosphere. Then angular momentum is lost from the star when the outer starlayers are removed. The envelope removal in massive stars that are progenitors of BBHmergers is easily accomplished either by stellar winds or by mass transfer to a closecompanion star. However, the efficiency of angular momentum transport is unknown.Two competitive models are currently considered in the literature: very effective angular


momentum transport by a magnetic dynamo [119, 120] included in the MESA stellarevolution code that leads to solid body rotation of the entire star, and mild angularmomentum transport through meridional currents [78, 121] included in the Geneva codethat leads to differential rotation of the star. Asteroseismology that probes the internalrotation of stars has not yet provided any data on massive stars (i.e. progenitors of BHs).The available measurements for intermediate-mass stars (B type main-sequence stars)show that some stars are well described by solid body rotation and some by differentialrotation [122]. Depending on an the adopted model, the angular momentum contentof a star at core-collapse could be very different. During BH formation some angularmomentum may be lost affecting the natal BH spin if material is ejected in a supernovaexplosion. Whether BH formation is accompanied by mass loss is not at all clear andestimates that use different assumptions on mass ejection in the core-collapse processare underway [123, 124]. At the moment, from the modeling perspective, the BH natalspin is mostly unconstrained.

4.1.2. Binary star evolution The majority (& 70%) of massive O/B stars, the potentialprogenitors of NSs and BHs, are found in close binary systems [125]. The evolution ofmassive stars in binaries deviates significantly from that of single stars [126–129]. Themain uncertainties affecting the calculation of BH merger rates are the metallicity, thecommon-envelope phase and the natal kick a BH receives at birth. These factors alsodetermine the two main BH properties: mass and spin.

Two main scenarios were proposed for BBH merger formation from stars thatevolve in galactic fields: classical isolated binary evolution similar to that developedfor double neutron stars (e.g., [27, 130–136]) and chemically homogeneous evolution(e.g., [93, 103, 137–140]). Classical binary evolution starts with two massive stars in awide orbit (a & 50 − 1000 R), and then binary components interact with each otherthrough mass transfers decreasing the orbit below ∼ 50 R in common envelope (CE)evolution [141, 142]. Depending on their mass, both stars collapse to BHs, either withor without supernova explosion, forming a compact BBH binary. The orbital separationof two BHs which merge within a Hubble time is below ∼ 50 R (for a circular orbitand two 30 M BHs [143]). [144] highlight that for the massive stars that are expectedto form BHs, the mass ratio in the second mass-transfer phase is much less extreme,which means a CE phase may be avoided.

In the chemically homogeneous evolution scenario, two massive stars in a low-metallicity environment form in a very close binary (. 50 R) and interact stronglythrough tides [145, 146]. Tidal interactions lock the stars in rapid rotation and allowfor the very effective mixing of elements in their stellar interior that inhibits radialexpansion of the stars. Hence, these stars remain compact throughout their evolutionand collapse to BHs without experiencing a CE phase [103]. This evolutionary schememay well explain the most massive LIGO/Virgo BBH mergers, as the enhanced tidalmixing required in this channel only works for most massive stars (& 30 M). It alsopredicts that both binary components evolve while rotating fairly rapidly and this may


produce rapidly spinning BHs, unless angular momentum is lost very efficiently in thelast phases of stellar evolution or during BH formation.

4.1.3. Reconciling observations and theory There seems to be some confusion in thecommunity as to what was expected and predicted by stellar/binary evolution modelsprior to LIGO/Virgo detections of the first sources. In particular, it is striking thatoften it is claimed that LIGO/Virgo detections of BBH mergers with very massive BHswere surprising or unexpected. Before 2010, in fact most models were indicating thatBNS are dominant GW sources for ground-based detectors (however, see also Ref. [133],predicting LIGO detection rates strongly dominated by BBH binaries), and that stellar-origin BHs are formed with small masses of ∼ 10 M [147]. The models before 2010were limited to calculations for stars with high metallicity (typical of the current MilkyWay population) and this has introduced a dramatic bias in predictions. However,already around 2010 it was shown that stars at low metallicities can produce much moremassive (30−80 M) BHs than observed in the Milky Way [96, 148, 149]. Additionally,it was demonstrated that binaries at low metallicities are much more likely to produceBBH mergers than high metallicity stars by one or two ordes of magnitude [25]. Thisled directly to the pre-detection predictions that (i) the first LIGO/Virgo detectionwas expected when the detector (BNS) sensitivity range reached about 50 − 100 Mpc(the first detection was made at 70 Mpc), that (ii) BBH mergers will be the firstdetected sources, and that (iii) the BBH merger chirp-mass distribution may reach30 M [25, 66, 150, 151]. Additionally studies of the future evolution of X-ray binarieslike IC10 X-1 and NGC300 X-1 [152] suggested that there exists a large population ofmerging BH binaries with masses in excess of 20 M.

Post-detection binary evolution studies expanded on earlier work to show agreementof calculated BBH merger rates and BBH masses with LIGO/Virgo observations [27,97, 135, 136]. The range of calculated merger rates (10− 300 Gpc−3 yr−1) comfortablycoincides with the observed rate estimates (12−213 Gpc−3 yr−1 for the LIGO/Virgo 90%credible interval). Note that these classical binary evolution rates are typically muchhigher than rates predicted for dynamical BBH formation channels (5− 10 Gpc−3 yr−1,[153, 154]). The most likely detection mass range that is predicted from classical isolatedbinary evolution is found in the total BBH merger mass range of 20−100 M (e.g.[97]).Examples of merger rate and mass predictions for BBH mergers are given in Figures 2and 3. A similar match between observed LIGO/Virgo BH masses and model predictionsis obtained from the dynamical formation channel [153, 154]. Note that this makes thesetwo channels indistinguishable at the moment, although the merger rates are likely tobe much smaller for the dynamical channel.

A caveat of concern for the prospects of LIGO/Virgo detecting BBH mergers withmasses above the PISN gap is related to the relatively low GW frequencies of the suchmassive BBH binaries with chirp masses above 100 M. During the in-spiral, theemitted frequencies are expected to peak approximately at the innermost stable circularorbit (ISCO), before the plunge-in phase and the actual merging. Hence, the emitted


0 20 40 60 80 100 120 140 160

O2 sensitivity(120 days)

LIGO rate:(arrow)

BH-BHonly

M20M10M26M25

M23M13

Figure 2. Left: Redshifted total merger mass distribution for two population synthesismodels [123]: M10 (low BH natal kicks) and M23 (high BH natal kicks). The O2LIGO sensitivity is marked; the most likely detections are expected when models areclosest to the sensitivity curve. We also mark LIGO/Virgo BBH merger detections(vertical positions have no meaning), all of which fall within the most likely detectionregion between 20− 100 M. Right: Source frame BBH merger-rate density of severalpopulation synthesis models for the local Universe (z = 0). The current LIGO O1/O2BBH merger rate is 12–213 Gpc−3 yr−1 (blue double-headed arrow). Note that themodels with fallback-attenuated BH natal kicks (M10, M20) are at the LIGO upperlimit, while models with high BH natal kicks are at the LIGO lower limit (M13, M23).Models with small (M26) and intermediate (M25) BH kicks fall near the middle of theLIGO estimate.

frequencies are most likely less than 100 Hz, and with redshift corrections the frequenciesto be detected are easily lower by a factor of two or more. A frequency this low is closeto the (seismic noise) edge of the detection window of LIGO/Virgo and may not beresolved.

The current LIGO/Virgo broad range of an empirically determined local BBHmerger-rate density (12 − 213 Gpc−3 yr−1) can easily be explained by uncertainties inkey input physics parameters of population synthesis modelling, such as the slope of theIMF or the efficiency of the CE ejection, see e.g. Table 5 in [97]. Alternatively, it may beexplained by altering BH natal kicks [27] from full NS natal kicks corresponding to a lowrate estimate) to almost no BH kicks (high rate estimate). Once LIGO/Virgo narrowsits empirical estimate it may be possible to use the merge-rate density to constrain theinput physics applied in modelling, although it should be cautioned that there is a largedegree of degeneracy [97, 133].

LIGO/Virgo provides an estimate of the effective spin parameter that measures theprojected BH spin components (a1, a2) parallel to binary angular momentum, weighted

Black holes, gravitational waves and fundamental physics: a roadmap 26Chirpmass,M

(M)

System mass, M (M)

< 4 · 10−11

10−10

10−9

10−8

10−7

> 3 · 10−7

GW

merger

rate

per

pixel

(yr−

1)

GW150914

LVT151012

GW170104GW170814

0

5

10

15

20

25

30

35

40

45

0 10 20 30 40 50 60 70 80 90 100

ZIZw18 = 0.0002

Figure 3. Distribution of simulated double compact object binaries in the total mass–chirp mass plane for a metallicity of Z = 0.0002. Three islands of data are visible,corresponding to BBH, mixed BH-NS and BNS systems. The colour code indicatesthe merger rate per pixel for a Milky Way equivalent galaxy. The three solid greylines indicate a constant mass ratio of 1, 3 and 10 (from top to bottom). ObservedLIGO/Virgo sources are shown with black crosses and event names are given for thefour most massive cases. The lowest mass BBH mergers can only be reproduced witha higher metallicity. Figure taken from Ref. [97].

by BH masses (M1,M2):

χeff ≡M1a1 cos Θ1 +M2a2 cos Θ2

M1 +M2, (3)

where Θ1,2 are the angles between the BH spins and the orbital angular momentumvector. So far, for the six LIGO/Virgo BBH detections the effective spins cluster around|χeff | < 0.35 (e.g., [11]). This defies the expectations for the main BBH formationchannels. If spin magnitudes of stellar-origin BHs are significant, as estimated forseveral Galactic and extra-galactic X-ray binaries [155], then the dynamical formationchannel (random BH capture) predicts an isotropic χeff distribution, while the classicalbinary evolution channel mostly predicts aligned BH spins (aligned stellar spins thatare only moderately misaligned by BH natal kicks). Hence, in the latter case oneexpects a distribution peaked at high values of χeff . On the one hand this tension is


rather unfortunate, as it does not allow to distinguish between these two very differentscenarios of BBH merger formation. On the other hand, this is a great opportunity tolearn something new about stars and BHs that was so far not expected and is not easilyunderstood in the framework of current knowledge.

There are five potential explanations of this apparent tension. First, there couldbe a mechanism that puts both BH spins in the plane of the binary orbit, producingχeff = 0 independent of BH masses and spin magnitudes. Such a mechanism is proposedto operate in triple stars [156]. Note that triple stars are a minority of stars (10−20% ofall field stars) and that the proposed mechanism requires very specific tuning to operate,so it is not clear how likely it is that it worked for all LIGO/Virgo sources. Second,there could be a mechanism that forces the BH spins to be in opposite directions so thatthey cancel out. For approximately equal mass BBH binaries (typical of LIGO/Virgosources) this would imply 180 degree flip of spins. No mechanism is known to producesuch configurations. Third, both BH spin magnitudes may be very small reducingeffective spin parameter to χeff = 0, independent of other merger parameters. Thiswas already proposed and is used in studies of angular momentum transport in stars[123]. Fourth, LIGO/Virgo BHs may not have been produced by stars, but for examplethey come from a primordial population for which small spins are naturally expected[74, 157]. Fifth, it may be the case that the spin of a BH (at least its direction) is notmainly determined by the angular momentum of the progenitor star, but a result ofthe physics of the collapse (spin tossing, e.g., [158]). In that case, there is no reasonto assume the spins in mergers formed from isolated binary evolution are aligned andthey may be isotropic. Note that with these five options, that need to be tested anddeveloped further, one cannot determine the main formation channel of BBH mergers,as it was proposed in several recent studies [146, 159–161].

The issue of spins is rather fundamental as the effective spin parameter most likelycontains information on natal BH spin magnitudes and therefore information on stellarastrophysics regarding angular momentum transport in massive stars, which is stillunconstrained by electromagnetic observations. Possibly the second-formed BH spincould have been increased in binary evolution by accretion of mass from a companionstar. However, it is argued that BHs cannot accrete a significant amount of mass inthe binary evolution leading to the BBH formation [27, 123, 162]. This is partly dueto very low accretion rates during a CE of 1 − 10% of the Bondi-Hoyle accretion rate[163–166]) and fast timescale Roche-lobe overflow (RLO) in massive progenitor binariesthat leads to ejection of most of the exchanged matter from the binary system (due tosuper-Eddington mass-transfer rates). The amount of mass accreted by BHs in binarysystems (. 1− 3 M) cannot significantly spin up massive BHs (10− 30 M) that aredetected by LIGO/Virgo.

It is important to note the most challenging parts of the evolutionary predictionsin the context of BBH formation. In the classical binary evolutionary channel, the twomost uncertain aspects of input physics are related to the CE evolution and the natalBH kicks. Although some observational constraints on both processes exist, they are


rather weak. Systems entering CE evolution have recently been reported. However,they are not as massive as stars that could produce NSs or BHs [167]. The search forCE traces as IR outburts has so far yielded no clear detection of emerging or recedingX-ray binaries as expected in this scenario [168].

BH natal kicks are only measured indirectly by positions and motions of X-raybinaries hosting BHs, and usually only lower limits on natal kicks are derived [115–117, 169]. On theoretical grounds, reliable models for CE [142] and supernovae [170]are missing. In the chemically homogeneous evolution channel the largest uncertainty isconnected with the efficiency of the mixing, the number of massive binaries that can formin very close orbits, and the strength of tidal interactions in close binaries. Since initialorbital period distributions are measured only in the very local Universe [125], it is notclear whether they apply to the majority of all stars and thus it is not fully understoodhow many stars are potentially subject to this kind of evolution. Even a deeper problemexists with our understanding of tides and their effectiveness in close binaries [145, 146],and effective tides are the main component of input physics in chemically homogeneousevolution.

Astrophysical inferences from GW observations are currently limited. First, itis not known which formation channel (or what mixture of them) produces the knownLIGO/Virgo BBH mergers. Since each channel is connected to specific set of conclusions(for example, the isolated binary channel informs about CE evolution and natal kicks;while the dynamical channel informs predominantly about stellar interactions in denseclusters) it is not clear which physics we are testing with GW observations. Second,within each channel there is degeneracy such that multiple model parameters areonly weakly constrained by observations. As nobody so far was able to deliver acomprehensive study of the large multi-dimensional parameter space, the inferences onvarious model parameters (e.g. the strength of BH natal kicks or the CE efficiency) arehindered by various and untested model parameter degeneracies. However, it is alreadypossible to test several aspects of stellar evolution as some processes leave unambiguoussignatures in GW observations. For example, the existence of the first and the secondmass gap, if confirmed by LIGO/Virgo, will constrain core-collapse supernovae andPISNe, respectively. Careful studies with detailed exposure of caveats are needed totransform future observations into astrophysical inferences.

It is expected that GW events resulting from the merger of stellar-mass BHs areunlikely to produce electromagnetic counterparts. Nevertheless, a (marginal) transientsignal detected by the Fermi gamma-ray burst monitor, 0.4 seconds after GW150914,was reported [171]. This claim encouraged several theoretical speculations for a possibleorigin. It has been suggested [172] that a tiny fraction of a solar mass of baryonicmaterial could be retained in a circumbinary disk around the progenitor binary whichprobably shed a total mass of > 10 M during its prior evolution. The sudden massloss and recoil of the merged BH may then shock and heat this material, causing atransient electromagnetic signal. It will be interesting see if any further electromagneticsignals will be associated with BBH mergers in the near future.


4.2. BNS mergers

The formation of double NSs has been studied in detail following the discovery ofthe Hulse-Taylor pulsar [173] and currently more than 20 such BNS systems are known– see [158] for a details and a review on their formation and evolution. Whereas no BBHbinaries had been detected prior to GW150914, detailed knowledge on BNS systems wasknown for many years from Galactic radio pulsar observations [174].

LIGO/Virgo has currently only detected one BNS merger event (GW170817),located in the lenticular (S0) galaxy NGC 4993, and thus the local empirical BNSmerger rate density still remains rather uncertain: 1540+3200

−1220 Gpc−3 yr−1 (90% crediblelimits [21]). The study of double NSs is relevant for the study of BHs because it givesindependent constraints on the evolution of similar massive binary populations fromwhich binary BHs are formed. In particular the question which stars for NSs and whichform BHs and if and how this depends on previous binary interactions is a questionthat likely only can be answered observationally by significant statistics on the relativeabundance of double BHs, BNS and NS-BH binaries.

There are two major sites to produce BNS mergers: isolated binaries in galacticfields (the main contributor), and dense environments in globular and nuclear clusters.None of these sites (nor any combination of them) can easily reproduce the preliminaryestimated LIGO/Virgo event rate, even if all elliptical host galaxies are included withinthe current LIGO/Virgo horizon [175]. The local supernova rate can be estimated tobe about 105 Gpc−3 yr−1, so the current empirical BNS merger rate from LIGO/Virgowould imply a very high efficiency of BNS binary formation.

This apparent tension may be solved if BNS mergers are allowed to originate from awide range of host galaxies and if the low-end of the LIGO/Virgo merger-rate estimateis used (320 Gpc−3 yr−1). Population synthesis studies seem to agree that rates as highas 200 − 600 Gpc−3 yr−1 can possibly be reached if favorable conditions are assumedfor classical binary evolution [97, 176, 177]. In the coming years, the statistics ofthe empirical BNS merger rate will improve significantly and reveal whether currenttheoretical BNS merger rates need a revision. It is interesting to notice, however, thatcalibrations with the rates of observed short gamma-ray bursts and the rate of mergersrequired to reproduce the abundances of heavy r-process elements favor a merger-ratedensity significantly smaller than the current empirical rate announced by LIGO/Virgo[97].

The main uncertainties of the theoretically predicted merger rate of BNS binariesare also related to CE evolution and SNe (similar to the case of BBH mergers). A CEevolution is needed to efficiently remove orbital angular momentum to tighten the binaryorbit and allow a merger event within a Hubble time. However, the onset criterion andthe efficiency of the in-spiral in a CE remain uncertain [142, 178]. The kick velocitiesimparted on newborn NSs span a wide range from a few km s−1 (almost symmetricSNe) to more than 1000 km s−1 and are sometimes difficult to determine [179]. The kickmagnitude seems to be related to the mass of the collapsing core, its density structure


and the amount of surrounding envelope material [111]. Additional important factorsfor the predicted merger rates include the slope of the initial-mass function and theefficiency of mass accretion during RLO [97].

All taken together, the predicted merger rate of double NSs in a Milky Wayequivalent galaxy vary by more than two orders of magnitude [147]. The empiricalmerger rate that LIGO/Virgo will detect at design sensitivity in a few years is ofuttermost importance for constraining the input physics behind the rate predictions.Besides the detection rates, the mass spectrum and the spin rates of the double NSmergers will reveal important information about their origin. Although their precisevalues cannot be determined due to degeneracy, the overall distribution of estimated NSmasses will reveal information on their formation process (electron capture vs iron-corecollapse SNe), as well as constraining the nuclear-matter equation-of-state. The latterwill also be constrained from tidal deformations of the NSs in their last few orbits [21].

An important observational signature of the merger event of BNS binaries is thedetection of the ring-down signal of either a meta-stable highly massive NS or a BHremnant. Such information would set constraints on the fundamental equation-of-stateof nuclear matter at high densities. Whereas LIGO/Virgo is not sensitive enough todetect a ring-down signal, it is the hope that third-generation GW detectors might beable to do so. Another important observational input is the distribution of mass ratiosin BNS merger events. This distribution could provide important information aboutthe formation of NSs and the nature of the supernovae (e.g. electron-capture vs ironcore-collapse supernovae).

Optical follow-up will in many cases reveal the location of a double NS merger(e.g., [22]). This will provide information on their formation environments [180, 181]and kinematics [182], besides crucial information on heavy r-process nucleosynthesis[183].

4.3. Mixed BH-NS mergers

The formation of mixed BH/NS mergers is expected to follow similar scenarios asdouble NS or double BH [130, 133] with all the associated uncertainties.

It is perhaps somewhat surprising that LIGO/Virgo detected a double NS merger(GW170817) before a mixed BH/NS merger, since (at least some) population synthesiscodes predict a detection rate of mixed BH/NS systems which is an order of magnitudelarger that the expected detection rate of double NS systems (with large uncertainties,e.g.,[97]). Hence, if these predictions are correct, GW170817 is a statistical rare eventand detections of mixed BH/NS systems are expected already in the upcoming O3/O4LIGO runs. The detection of mixed BH-NS mergers is interesting for two reasons:(i) a key question is whether BH-NS and NS-BH binaries may be distinguished fromone another (i.e. the formation order of the two compact objects, which leads to a(mildly) recycled pulsar in the latter case), and (ii) the detected in-spiral of mixed BH-NS mergers may reveal interesting deviations from GR and pure quadrupole radiation


given the difference in compactness between BHs and NSs [184].

5. Dynamical Formation of Stellar-mass Binary Black HolesContributors: B. Kocsis and A. Askar

5.1. Introduction

The recent GW observations from six BBH mergers (GW150914, LVT151012,GW151226, GW170104, GW170608, GW170814) and a BNS merger (GW170817)opened ways to test the astrophysical theories explaining the origin of these sources[1, 2, 11, 18, 20, 21] . As discussed earlier, the component masses of these merging sourcesspan a range between 8–35M [11], which is different from the distribution of BHs seenin X-ray binaries, 5 – 17M [185] with two possible exceptions (NGC300X-1 and IC10X-1). The event rates of BBH mergers are estimated to be between 40–213 Gpc−3yr−1 fora power-law BH mass function and between 12–65 Gpc−3yr−1 for a uniform-in-log BHmass function [11], which is higher than previous theoretical expectations of dynamicallyformed mergers, for instance see [147]. The event rates of BNS mergers is currently basedon a single measurement which suggests a very high value of 1540+3200

−1220 Gpc−3yr−1 [21](c.f. [175]). How do we explain the observed event rates and the distribution of masses,mass ratios, and spins?

Several astrophysical merger channels have been proposed to explain observations.Here we review some of the recent findings related to dynamics, their limitations anddirections for future development. These ideas represent alternatives to the classicalbinary evolution picture, in which the stars undergo poorly understood processes, such ascommon envelope evolution. In all of these models the separation between the compactobjects is reduced dynamically to less than an AU, so that GWs may drive the objectsto merge within a Hubble time, tHubble = 1010 yr.

5.2. Merger rate estimates in dynamical channels

Dynamical formation and mergers in globular clusters Although about 0.25% ofthe stellar mass is currently locked in globular clusters (GCs) [186–188], dynamicalencounters greatly catalyze the probability of mergers compared to that in the field.Within the first few million years of GC evolution, BHs become the most massive objects.Due to dynamical friction, they will efficiently segregate to the cluster center [189]where they can dynamically interact and form binaries with other BHs [190, 191]. Thedense environments of GCs can also lead to binary-single and binary-binary encountersinvolving BHs that could result in their merger. Collisional systems like GCs can alsoundergo core collapse, during which central densities can become very large leadingto many strong dynamical interactions. The encounter rate density is proportional toR ∼

∫dV 〈n2

∗〉σcsv, where n∗ is the stellar number density, σcs ∼ GMb/v2 is the capturecross section, M is the total mass, b is the impact parameter, v is the typical velocity


dispersion. Note the scaling with 〈n2∗〉, where 〈n2

∗〉1/2 ∼ 105pc−3 in GCs and ∼ 1pc−3 inthe field.

Estimates using Monte Carlo method to simulate realistic GCs yield merger rates ofat least RGC ∼ 5 Gpc−3yr−1 [153, 154], falling below the current limits on the observedrates. Rate estimates from results of direct N–body simulations also yield a similar valueofRGC ∼ 6.5 Gpc−3yr−1 [192]. In particular, these papers have shown that the low-massGCs below 105M have a negligible contribution to the rates. However, they also showthat initially more massive GCs (more massive than 106M) contribute significantly tothe rates. [154] argue that actual merger rates from BHs originating in GCs could be3 to 5 times larger than their estimated value of ∼ 5 Gpc−3yr−1 due to uncertaintiesin initial GC mass function, initial mass function of stars in GCs, maximum initialstellar mass and evolution of BH progenitors. Furthermore, BBH merger rates can besignificant in young clusters with masses ∼ 104 M [193, 194].

A simple robust upper limit may be derived by assuming that all BHs merge oncein each GC in a Hubble time:

R ≤ 12fBHN∗

nGC

tHubble

<12

∫ 150M20M fIMF(m)dm∫ 150M

0.08M mfIMF(m)dm× 105.5M ×

0.8Mpc−3

1010yr= 38 Gpc−3yr−1 (4)

where fBH is the fraction of stars that turn into BHs from a given stellar initial massfunction fIMF, N∗ is the initial number of stars in a GC, and nGC is the cosmic numberdensity of GCs, and fIMF(m) ∝ (m/0.5M)−2.3 for m > 0.5M and (m/0.5M)−1.3

otherwise [195]. The result is not sensitive to the assumed upper bound on mass of theBH progenitor, which is set by the pair instability supernova. Recent estimates find110–130 M for GC metallicities [31]. However, note that the mass in GCs may havebeen much higher than currently by a factor ∼ 5, since many GCs evaporated or gottidally disrupted [196, 197]. This effect increases the rates by at most a factor 2 atz < 0.3, but more than a factor 10 at z > 2.5 [198].

Dynamical and relativistic mergers in galactic nuclei The densest regions of stellarBHs in the Universe are expected to be in the centers of nuclear stars clusters (NSC),whose mass-segregated inner regions around the SMBH exceed n∗ ∼ 1010pc−3 [199]. Incontrast to GCs, the escape velocity of the SMBH in the central regions of NSCs isso high that compact objects are not expected to be ejected by dynamical encountersor supernova birth kicks. In these regions, close BH binaries may form due to GWemission in close single-single encounters [199]. Binary mergers may also be induced bythe secular Kozai-Lidov effect of the SMBH [200–205] and tidal effects [206]. Detailedestimates give from the RNSC ∼ 5 − 15 Gpc−3yr [175, 199–202], below the observedvalue. Higher values may be possible for top heavy BH mass functions and if blackholes are distributed in disk configurations [199, 207].


These numbers are sensitive to the uncertain assumptions on the total supply ofBHs in the NSCs, either formed in situ or delivered by infalling GCs [196, 208–211].If all BHs merged in galactic nuclei once, an upper limit similar to Equation (4) isRNSC < 30 Gpc−3yr−1. Due to the high escape velocity from NSCs and a rate ofinfalling objects, this bound may be in principle exceeded.

Mergers facilitated by ambient gas Dynamical friction facilitates mergers in regionswhere a significant amount of gas embeds the binary, which may carry away angularmomentum efficiently. Particularly, this may happen in a star forming regions[75, 212, 213], in accretion disks around SMBHs in active galactic nuclei (AGN) [214, 215]or if the stellar envelope is ejected in a stellar binary [216]. In AGN, the accretion diskmay serve to capture stellar mass binaries from the NSC [217], or help to form them inits outskirts by fragmentation [218]. The rate estimates are at the order of 1 Gpc−3yr−1,below the observed value. Nevertheless, mergers in this channel deserve attention asthey have electromagnetic counterparts, the population of AGNs, which may be used ina statistical sense [219].

Isolated triples The stellar progenitors of BHs are massive stars, which mostly residenot only in binaries, but in many cases in triples. The gravity of the triple companiondrives eccentricity oscillations through the Kozai-Lidov effect, which may lead to GWmergers after close encounters. However, the rate estimates are around or below6 Gpc−3yr−1, below the current observational range [220, 221]. The rates may be higher2–25 Gpc−3yr−1 for low metallicity triples [222] and further increased by non-hierarchicalconfigurations [223, 224] and by quadrupole systems [225].

Mergers in dark matter halos The first metal-free stars (Pop III) may form binariesdynamically and merge in DM halos [226], but the expected rates are below the observedrates [75]. Futhermore, two dynamical channels have been suggested leading to a highnumber of BH mergers in DM halos. If PBHs constitute a significant fraction of DM,the merger rates following GW captures in single-single encounters match the observedrates [227]. However, this would also lead to the dispersion of weakly bound GCs inultrafaint dwarf galaxies, contradicting an observed system [228]. More recent estimatesshow that the LIGO rates are matched by this channel even if only 1% of the DM is inPBHs [229]. The second PBH channel requires only a 0.1%– fraction of DM to be PBHs,given that they form binaries dynamically in the very early universe [230]. While thesesources can match the observed rates, as discussed in the following Sec. 6, we awaitfurther strong theoretical arguments or observational evidence for the existence of thesePBHs [231].


5.3. Advances in numerical methods in dynamical modeling

Recent years have brought significant advances in modelling the dynamicalenvironments leading to GW events.

5.3.1. Direct N-body integration State of the art direct N–body simulations have beenused to model the dynamics of GCs to interpret GW observations. These methodsnow reach N = 106, so that stars are represented close to 1:1 in the cluster [232].Comparisons between the largest direct N–body and Monte Carlo simulations showan agreement [232, 233]. However, due to their high numerical costs, only a very lownumber of initial conditions have been examined to date. Further development is neededto account for a higher number of primordial binaries, larger initial densities, a realisticmass spectrum, and a nonzero level of rotation and anisotropy.

Large direct N -body simulation have also been used to study the dynamics innuclear star clusters (NSC) in galactic nuclei with an SMBH and the formation of NSCsfrom the infall of GCs [210]. Recent simulations have reached N = 106 in regionsembedding NSCs [234]. Here, the number of simulated stars to real stars is 1:100. Tointerpret GW mergers in these systems, a 1:1 simulation of the innermost region of theNSC would be most valuable, even if the total simulated number is of the order 105–106.Including a mass distribution and primordial binaries would also be useful.

Direct N -body methods were also recently used to simulate the NSC in AGN withstellar captures by the accretion disk, to predict the rate of tidal disruption eventsand the formation of a nuclear stellar disk [235, 236]. The most important place fordevelopment is to relax the assumption of a rigid fixed disk in these simulations. Indeed,the disk may be expected to be significantly warped by the star cluster [237, 238], andthe stars and BHs captured in the disk may grow into IMBHs and open gaps in the disk[215]. Furthermore, an initially nonzero number of binaries, the binary-disk interaction,and a mass spectrum would be important to incorporate to make GW predictions.

5.3.2. Monte-Carlo methods State of the art Monte-Carlo methods have also improvedsignificantly during the past years, providing a numerically efficient way to model theevolution of GCs accurately. Recent developments showed that the BH subclusters donot necessarily decouple and evaporate at short time scales and that GCs with longerrelaxation times can retain BHs up to a Hubble time [239–241]. These methods havebeen used to predict the evolutionary pathways to some of the observed mergers [69, 242]and to interpret the distributions of masses and spins [153, 154]. These methods havebeen used to study the formation of IMBHs [243]. Most recently, 2.5-order post-Newtonian dissipative effects were incorporated in order to re-simulate binary-singleinteractions involving three BHs from results of these Monte Carlo codes which increasedthe rate of eccentric mergers by a factor of 100 [244–246]. The implementation of post-Newtonian terms and tidal dissipation [247] for computing results of strong binary-singleand binary-binary encounters involving at least two BHs could further increase merger


rates for BBHs expected to originate from GCs. Moreover, implementation of two bodygravitational and tidal capture within these codes could provide more insights into therole of dynamics in forming potentially merging BBHs.

Further development is needed to include resonant multibody effects. Moreover,simulations of galactic nuclei with Monte Carlo methods would be valuable to trackingthe evolution of binaries, and accounting for long term global effects such as resonantrelaxation.

5.3.3. Secular Symplectic N-body integration Systems which are described by aspherical gravitational potential such as a galactic nucleus or a GC are affected bystrong global resonances, in which the anisotropies of the system drive a rapid secularchange in the angular momentum vectors of the objects, called resonant relaxation [248].Vector and scalar resonant relaxation, which affect the distribution of orbital planes andthe eccentricity, respectively, are expected to reach statistical equilibrium within a fewMyr to a few 100 Myr, respectively. A secular symplectic N -body integration methodwas recently developed [249]. Preliminary results show that objects such as BHs, whichare heavier than the average star in a GC or a nuclear star cluster tend to be distributedin a disk [207]. Since LVC mergers happen more easily in BH disks, if they exist, futurestudies are necessary to explore the formation, evolution, and the expected propertiesof such configurations.

The statistical equilibrium phase space distribution of resonant relaxation is knownonly for a limited number of idealized configurations [250–253]. Interestingly resonantrelaxation has strong similarities to other systems in condensed matter physics such aspoint vortices and liquid crystals [249–251]. This multidisciplinary connection may beused to study these probable sites of BH mergers [251].

5.3.4. Semianalytical methods Semianalytical methods are developed to include thehighest number of physical effects in an approximate way. The formation of the Galacticbulge from the infall of GCs was examined in great detail with this technique [196, 197].It was shown, that during this process, more than 95% of the initial GC population isdestroyed. Thus, GCs were much more common at cosmologically early times. Since oneway to form IMBHs in GCs is by runaway collisions of stars or BHs [254], their numbersmay also expected to be higher than previously thought (however, this conclusion maybe different in other IMBH formation channels [243]). GCs generate a high rate ofmergers between stellar mass and 102–103M IMBHs detectable in the near future withadvanced LIGO and Virgo detectors at design sensitivity at z > 0.6 [203, 204, 255]. BHmergers with IMBHs with 103–104M may also be detected with LISA from the localUniverse.


5.4. Astrophysical interpretation of dynamical sources

How can dynamical models test the astrophysical interpretation of GW sources?GW detectors can measure the binary component masses, the spin magnitudes anddirection, the binary orbital plane orientation, the eccentricity, the distance to thesource and the sky location. The observed distributions of these parameters may becompared statistically to the predicted distributions for each channel. Some smokinggun signatures of the astrophysical environment are also known for individual sources,discussed below.

5.4.1. Mass distribution The mass distribution of mergers depends on the theoreticallypoorly known BH mass function times the mass-dependent likelihood of mergers.Particularly, the mergers of massive objects are favored in GCs, due to the massdependence of binary formation in triple encounters, binary exchange interactions,dynamical friction, and the Spitzer instability. Monte Carlo and N -body simulationsshow that the likelihood of merger is proportional toM4 in GCs [256]. The 2-dimensionaltotal mass and mass ratio distribution of mergers may be used to test this prediction[257] and to determine the underlying BH mass function in these environments. Themass function of mergers may also vary with redshift as the most massive BHs areejected earlier [153].

Recently, Ref. [258] introduced a parameter to discriminate among differentastrophysical channels:

α = −(m1 +m2)2 ∂2R∂m1∂m2

. (5)

This dimensionless number is 1±0.03 for PBHs formed dynamically in the early universe(if the PBH mass function is assumed to be flat. For arbitrary PBH mass function theresults can be substantially different [259]), 10/7 for BHs which form by GW emissionin collisionless systems such as DM halos. For BHs which form by GW emission incollisional systems which exhibit mass segregation, this α varies for different componentmasses. In galactic nuclei it ranges between 10/7 for the low mass components in thepopulation to −5 for the highest mass component. It would be very useful to makepredictions for α for all other merger channels.

5.4.2. Spin distribution Using the empirically measured rotation rate of Wolf-Rayet(WR) stars, the birth spin of the massive BHs is expected to be small [260]. If BHs haveundergone previous mergers, their spin is distributed around 0.7 [53, 261]. Dynamicaleffects may also spin up the WR-descendent BH [159]. If the BH acquires a significantamount of mass due to accretion, it is expected to be highly spinning. Thus, secondgeneration mergers are distinguishable in mass and spin from first generation mergers.Monte Carlo simulations predict that 10% are second generation in GCs [244, 256].Results from Monte Carlo simulations have also been utilized to investigate the role ofspin in gravitational recoil kicks on merged BHs [262]. This has important implications


for repeated mergers in dense environments like GCs, results from Ref. [262] suggest thatthat about 30% of merging BHs could be retained in GCs. According to a recent X-rayobserving campaign, 7 out of the 22 Active Galactic Nuclei analyzed are candidates forbeing high spin SMBHs (with spin > 98%), see Table 1 of Ref. [263]. Some of the X-raybinaries show evidence of highly spinning stellar mass BHs [264]. However, with theexception of a single source, current LVC sources are consistent with zero effective spin[265]. If spinning, the relative direction of spins is expected to be uncorrelated (isotropic)for spherical dynamical systems. This is different from the standard common envelopechannel, where spin alignment is generally expected with the angular momentum vectorand counteralignment is not likely, in case the BHs are spinning [265, 266].

5.4.3. Eccentricity distribution Since GW emission circularizes binaries [143], theyare expected to be nearly circular close to merger unless they form with a very smallpericenter separation. Indeed, GW sources in GCs are expected to have a relativelysmall eccentricity in the LIGO band close to merger [267]. Moderate eccentricity ofe = 0.1 is expected from 10% of GC sources at the low-frequency edge of the AdvancedLIGO design sensitivity [244, 245]. However, they may have a high eccentricity in theLISA band [246, 267–270] or the DeciHz band [271]. Field triples may also yield someeccentric LIGO mergers [220–223]. Eccentricity may be much higher for sources formingby GW emission in close encounters [199].

The eccentricity distribution of merging BH binaries in GCs is expected to havethree distinct peaks corresponding to binaries which merge outside of the cluster afterejection following a binary-single hardening interaction, binary mergers which happenwithin a cluster in between two binary-single interactions, and mergers which happenduring a binary-single interactions [246, 269].

For GW capture binaries, the typical eccentricity at the last stable orbit (LSO,extrapolating [143]) is set by the velocity dispersion of the source population σ aseLSO ∼ 0.03(σ/1000kms−1)35/32 (4η)−35/16, where η = m1m2/(m1+m2)2 is the symmetricmass ratio [272]. The heavier stellar BHs and IMBHs are expected to merge with a highereccentricity at around 0.1, while the lower mass BHs will have an eccentricity around10−3 [272]. At design sensitivity Advanced LIGO and VIRGO may be more sensitive toeccentricity well before merger [273].

5.4.4. Sky location distribution Since GW sources are observed from beyond 100 Mpc,they are expected to be isotropically distributed. The sky location measurementaccuracy is typically insufficient to identify a unique host galaxy counterpart forindividual mergers. However, the power spectrum of sky location of all mergers maybe measured. This may be useful to determine the typical galaxy types of mergers,particularly, whether the sources are in active galactic nuclei [219].

5.4.5. Smoking gun signatures Which other features may help to conclusively identifythe host environment of individual GW sources?


Resolved clusters with electromagnetic counterparts Recently it was shown that severalinspiraling stellar mass black hole binaries of Milky Way GCs are expected to be inthe LISA band [274]. Therefore if LISA identifies these GW sources on the sky, it willconstrain the event rates corresponding to the GC channel.

Modulation of the GW phase due to environmental effects In the case of LISA sources,extreme mass ratio inspirals may be significantly affected by an a SMBH companion orgas-driven migration [215, 275]. For stellar-mass BH mergers, the most important effectof a perturbing object is a Doppler phase shift, which accumulates mostly at low GWfrequencies [276]. LIGO and LISA together will be able to measure the SMBH providedthat the orbital period around the SMBH is less than O(yr) [276, 277].

GW astrophysical echos The identification of a secondary lensed signal by the SMBHusing LVC may confirm that a LVC merger takes place in a galactic nucleus. Thismanifests as a GW echo with a nearly identical chirp waveform as the primary signalbut with a typically fainter amplitude depending on direction and distance from theSMBH [278]. If the distance between the source and the SMBH is less than 103MSMBH,the relative amplitude of the echo is of the order of 10%, and the time-delay is less thana few hours. Studies of the expected source parameters of sources with GW echos areunderway.

Double mergers and mergers with electromagnetic counterparts Finally, a case isconceivable in which in which a binary-single or binary-binary encounter results in adouble merger, i.e. two mergers from the same direction within a short timespan [279].Such an observation would indicate a dense dynamical host environment. Furthermore,binary-single or binary-binary encounters involving at least 2 BHs and one or more stellarobject may also lead to BBH mergers with with transient electromagnetic counterpartsassociated with the disruption or accretion of stellar matter on the BHs. Observationof such a counterpart would also indicate that the merging BHs originated in a densedynamical environment. The inclusion of tidal and gravitational dissipation effectsduring the computation of such strong encounters [247] within simulations of GCs couldhelp to constrain rates for BBH mergers in which an electromagnetic counterpart couldbe expected.

6. Primordial Black Holes and Dark MatterContributors: G. Bertone, C. T. Byrnes, D. Gaggero, J. García-Bellido,

B. J. Kavanagh

6.1. Motivation and Formation Scenarios.

The nature of Dark Matter (DM) is one of the most pressing open problems ofModern Cosmology. The evidence for this mysterious form of matter started to grow in


the early 20th century [280], and it is today firmly established thanks to a wide arrayof independent observations [281]. Its nature is still unknown, and the candidates thathave been proposed to explain its nature span over 90 orders of magnitude in mass, fromultra-light bosons, to massive BHs [282–284]. An intriguing possibility is that at leasta fraction of DM is in the form of PBHs [285]. The recent LVC detections [227, 286],have revived the interest in these objects, and prompted a reanalysis of the existingbounds [285, 287] and new prospects for phenomenological signatures [157, 288].

PBHs might be produced from early universe phase transitions, topological defects(e.g., cosmic strings, domain walls), condensate fragmentation, bubble nucleation andlarge amplitude small scale fluctuations produced during inflation (see the reviews [231,289] and references therein). Excluding the possibility of BH relics, non-evaporatingPBHs could range in mass from 10−18 to 106 solar masses today, although the mostinteresting range today is that associated with the observed LIGO BBHs, around afew to tens of solar masses. For those PBHs, amongst the most promising productionmechanisms was the one first proposed in [290] from high peaks in the matter powerspectrum generated during inflation. When those fluctuations re-enter the horizonduring the radiation era, their gradients induce a gravitational collapse that cannotbe overcome even by the radiation pressure of the expanding plasma, producing BHswith a mass of order the horizon mass [291]. Most of these PBH survive until today,and may dominate the Universe expansion at matter-radiation equality.

One important question is what fraction fPBH = ΩPBH/ΩDM of DM should be madeof O(10M) PBHs in order to match the BBH merger rate inferred by LIGO/Virgo(R ' 10 - 200 Gpc−3 yr−1 [11]). If one considers PBH binaries that form withinvirialized structures, the corresponding rate is compatible with fPBH = 1 [227]. However,PBH binary systems can form in the early Universe (deep in the radiation era) as well[292, 293], if the orbital parameters of the pair allow the gravitational pull to overcomethe Hubble flow and decouple from it. A recent calculation of the associated mergerrate today [230] – significantly extended in [229] – provides a much larger estimate(compared to the former scenario): This result can be translated into a bound on fPBH,which is potentially stronger than any other astrophysical and cosmological constraintin the same range. The constraint has been recently put on more solid grounds in [294]by taking into account the impact of the DM mini-halos around PBHs, which have adramatic effect on the orbits of PBH binaries, but surprisingly subtle effect on theirmerger rate today. Several aspects of this calculation are still under debate, includingthe PBH mass distribution, the role of a circumbinary accretion disk of baryons, theimpact of initial clustering of PBHs (see [295]) and the survival until present time ofthe binary systems that decoupled in the radiation era.

The critical collapse threshold to form a PBH is typically taken to be δc ≡δρc/ρ ∼ 0.5, with an exponential sensitivity on the background equation of state, theinitial density profile and angular momentum. Because any initial angular momentumsuppresses PBH formation, PBHs are expected to spin slowly, a potential way todiscriminate between them and (rapidly rotating) astrophyical BHs [296].


10−2 10−1 100 101 102 103 104 105

MPBH [M]

10−3

10−2

10−1

100

DM

frac

tion

f PB

H=

ΩP

BH/Ω

DM

EROS+MACHO

Eridanus II

Accretion - radio

Accretion - X-ray

CMB - PLANCK

CMB - FIRAS

Figure 4. Summary of astrophysical constraints on PBHs in the mass rangeM ∈ [10−2, 105]M. Details of the constraints are given in the main text and we plothere the most conservative. We emphasize that astrophysical constraints may havesubstantial systematic uncertainties and that the constraints shown here apply onlyfor monochromatic mass functions.

Inflationary models which generate PBHs are required to generate a much largeramplitude of perturbations on small scales compared to those observed on CMB scales.This can be achieved either through multiple-field effects or an inflection point in single-field inflation [297, 298]. These typically generate a reasonably broad mass fractionof PBHs, in contrast to the monochromatic mass spectrum usually assumed wheninterpreting observational constraints. In addition, the softening of the equation ofstate during the QCD phase transition greatly enhances the formation probability ofsolar mass PBHs compared to the more massive merging BHs LIGO detected [299]. BHsbelow the Chandrasekhar mass limit would be a smoking gun of a primordial origin.

At non-linear order, scalar and tensor perturbations couple, implying that the largeamplitude perturbations required to generate PBHs will also generate a stochasticbackground of GWs. For the LIGO mass range the corresponding GW frequencyis constrained by pulsar timing arrays unless the scalar perturbations are non-Gaussian [300, 301].

6.2. Astrophysical probes

Constraints on the PBH abundance are typically phrased in terms of fPBH =ΩPBH/ΩDM, the fraction of the total DM density which can be contributed by PBHs.Even in the case where fPBH < 1, a future detection via astrophysical probes or GW


searches remains hopeful [230]. We outline selected astrophysical constraints below,summarizing these in Fig. 4, where we focus on PBHs around the Solar mass range,most relevant for GW signals.

Micro-lensing: The MACHO [302] and EROS [303] collaborations searched formicro-lensing events in the Magellanic Clouds in order to constrain the presence ofMassive Compact Halo Objects (MACHOs) in the Milky Way Halo. Consideringevents on timescales of O(1-800) days constrains fPBH < 1 for masses in the rangeMPBH ∈ [10−7, 30]M (though these constraints come with a number of caveats, see e.g.[287, 304]). Another promising target is M31: an important fraction of the Andromedaand Milky Way DM halo can be probed by micro-lensing surveys, and an interestinghint comes from the observation of 56 events in the O(1-100) days range from thisregion of interest, which may suggest a MACHO population with f ∼ 0.5 and mass1M or lighter [305, 306]. A recent search for lensing of Type Ia Supernovae obtainedconstraints on all PBH masses larger than around 10−2M, although substantial PBHfractions fPBH . 0.6 are still compatible with the data [307]. For wide mass distributionsthe SN lensing constraints go away [74] and PBH could still constitute all of the DM.

Early Universe constraints: PBHs are expected to accrete gas in the Early Universe,emitting radiation and injecting energy into the primordial plasma. This in turn can leadto observable distortions of the CMB spectrum and can affect CMB anisotropies [308].Early calculations over-estimated the effect [309, 310], with more recent studies findingweaker constraints [311, 312]. In spite of this, data from COBE/FIRAS [311, 313] andPLANCK [312] can still rule out PBHs with masses MPBH & 100M as the dominantDM component. It should be noted that these constraints depend sensitively on thedetails of accretion onto PBHs in the Early Universe (see e.g. Ref. [314]).

Dynamical constraints: The presence of PBHs is also expected to disrupt thedynamics of stars. Wide halo binaries may be perturbed by PBHs and the actualobservation of such systems constrains the PBHs abundance above ' 20M [315](although significant fractions fPBH . 0.2 are still allowed). PBHs are also expected todynamically heat and thereby deplete stars in the centre of dwarf galaxies. Observationsof stellar clusters in Eridanus II [228] and Segue I [316] have been used to constrain PBHsheavier than O(1)M to be sub-dominant DM components, unless they have an IMBHat their center [317].

Radio and X-ray constraints: If PBHs exist in the inner part of the Galaxy, whichcontains high gas densities, a significant fraction of them would inevitably form anaccretion disk and emit a broad-band spectrum of radiation. A comparison with existingcatalogs of radio and X-ray sources in the Galactic Ridge region already rules out thepossibility that PBHs constitute all of the DM in the Galaxy, even under conservativeassumptions on the physics of accretion [318]. During the next decade, the SKAexperiment will provide an unprecedented increase in sensitivity in the radio band;in particular, SKA1-MID will have the unique opportunity to probe the presence of asubdominant population of PBHs in the Galaxy in the 10÷ 100M mass range, even ifit amounts to a fraction as low as ' 1% of the DM.


In Fig. 4, we show only the most conservative of these constraints, which suggestthat PBHs may still constitute all of the DM for masses close to 10M, or a smallerfraction (fPBH & 0.1) for masses up to O(100M). We highlight that such astrophysicalconstraints may have large systematic uncertainties and that these constraints applyonly to PBHs with a mono-chromatic mass function. Recent studies have begun re-evaluating these constraints for extended mass functions [287, 319–321]. For physicallymotivated mass functions, it may be possible to achieve up to fPBH ∼ 0.1 for PBHs in themass range 25− 100M [322]. Relaxing certain dynamical constraints (which typicallyhave large uncertainties) or considering more general mass functions may accommodatean even larger PBH fraction [322, 323].

6.3. Discriminating PBHs from astrophysical BHs

A number of observations may help discriminating PBHs from ordinaryastrophysical BHs. The detection of GWs from a binary system including a compactobject lighter than standard stellar BHs, say below 1 solar mass, would point towardsthe existence of PBHs. This can be deduced from the highest frequency reached in theGW chirp signal, fISCO = 4400 Hz (M/M), and it is in principle possible already withAdvanced LVC in the next run O3 [288]. The detection of GWs at redshift z & 40would imply a non-standard cosmology, or the existence of PBHs [324]. Further insightson the origin of BHs might be obtained through the analysis of ‘environmental’ effects,which are discussed in Sec. 5, and through the analysis of the spatial distribution andmass function of X-ray and radio sources powered by BHs.

7. Formation of supermassive black hole binaries in galaxy mergersContributors: M. Colpi, M. Volonteri, A. Sesana

When two galaxies, each hosting a central SMBH merge, the SMBHs start a longjourney that should bring them from separations of tens of kpc (1 kpc = 3.086×1021 cm)down to milli-parsec scales, below which they merge through GWs. The initial pairing ofBHs in merging galaxies, their binding into a binary on parsec scales, and their crossingto enter the GW-driven inspiral are the three main stages of this dynamical journey,sketched in Figure 5 [65]. In some cases, the two SMBHs become bound in the core ofthe merger remnant, cross to the gravitational-driven regime and eventually coalesce.However, there are cases in which the two SMBHs never form a binary: one of theSMBHs may remain stranded and unable to pair and bind [325].

The cosmic context, the growth of DM halos through mergers with other halosgives us the backdrop upon which all the subsequent evolution develops. This is thefirst clock: the halo merger rate evolves over cosmic time and peaks at different redshiftfor different halo masses. Furthermore, the cosmic merger rate predicts that not allmergers occur with the same frequency: major mergers, of halos of comparable masses(from 1:1 down to ∼ 1:4) are rare, while minor mergers (mass ratio < 1 : 10) are more


II. BINARY FORMATION &

HARDENING III. GW DOMAIN I. PAIRING

log

t

tH

HUBBLE TIME

1-10 kpc

0.001 pc 1 pc

1 Gyr

THE LONG JOURNEY TRAVELLED BY MASSIVE BLACK HOLES IN MAJOR MERGERS

BLACK HOLE SEPARATION

micro-pc

scattering off single stars

MBH dynamics: binary formation 5

Figure 3. Face–on gas density map for run ‘ThFBl’ at timet = 2.1 Myr (i.e., when the star formation rate is maximum,see Fig. 2). The gas shocked after the first disc collision frag-ments into a large number of small clumps which very rapidlyconvert gas into new stellar mass. The black dots correspond tothe positions of the two MBHs.

clump distance when the transient binary system forms is 10 20 pc, which is always resolved with a number ofcells > 10, thanks to the refinement prescription describedin Lupi, Haardt & Dotti (2015), allowing us to accuratelyresolve the BH–clump close interaction.

In the case of run ‘ThFBh’, because of the relativelyhigher density threshold for SF, a slightly larger number ofmore massive clumps forms, resulting in the more disturbedorbits (and faster decay) seen in Fig. 1.

We have then compared the above analysis regardingthe MBH dynamics with runs employing the aforementionedblast wave–like feedback from SNe (BWFB–like runs). Asdiscussed before, this feedback implementation aims at de-scribing non–thermal processes in the aftermath of SNa ex-plosions. We find that the dynamical evolution of the MBHsis largely independent upon the details of the SNa feed-back employed, making MBH dynamics results fairly robustagainst the di↵erent implementations of sub–grid physics.

3.2 Gas dynamics

We discuss here the dynamics of the gas during the mergerevent. We focus on the case with the low–density thresholdfor SF (run ‘ThFBl’), keeping in mind that the higher den-sity case produces a qualitatively and quantitatively similaroutcome.

Fig. 6 shows the gas distribution around the MBHB af-ter t = 11 Myr. On large scale (left–hand panel), the relicdisc resulting from the collision of the progenitor discs is al-most totally disrupted because of SNa feedback. This resid-ual structure is counter–rotating relative to the MBHB or-bit. On scales of the order of few pc (right–hand panel), thegas which has not been converted into stellar particles set-tles in a circumbinary disc, with a total mass of few 105 M.The small disc corotates with the MBHB thanks to the drag-ging of gas by the MBHs during their inspiral towards thecentre. Note that this implies that the angular momentum

Figure 4. Total mass in clumps for run ‘ThFBl’. The red dia-monds correspond to the times at which we computed the clumpmass distribution shown in Fig. 5.

of the residual gas changed sign during the evolution of thesystem.

We report in Fig. 7 the evolution of the modulus ofMBH orbital angular momentum and compared it to themodulus of the total angular momentum of the gas whichis the closest to the MBHs in the simulation, defined asthe gas within a sphere of radius equal to 0.5 times theMBH separation. We observe that at the beginning of thesimulation the angular momentum of the gas is larger thanthat of the MBHs, and we remind that the gas is counter–rotating. After > 4 Myr, the angular momentum associatedwith the MBH orbit exceeds that of the gas and in principlethere are the conditions for a change in the sign of the gasangular momentum, being dragged by the MBHs. The gasangular momentum actually changes sign after 9 Myr,when the MBH separation is 45 pc. At this evolutionarystage, a large fraction (> 90%) of the initial gas mass isalready converted in stellar particles. After ' 10 Myr, whenSNe start to explode, the released energy is radiated awayby the small amount of residual gas, which is however unableto form further stellar mass at a comparable rate. In otherwords, star formation is not halted by SNa feedback, ratherby gas consumption.

Concerning the impact of blast wave feedback (BWFB–type runs), as expected it does not alter the gas dynamicsfor a time tSN (at that point the two MBHs have alreadyreached the centre of the system). After that time, the al-most simultaneous SNa events release a fairly large amountof energy which heats the gas up but is not radiated away.The net result is that the remaining gas is pushed at verylarge distances from the MBHB (up to 500 pc) by theincreased pressure. The MBHB lives then in a very low–density environment, and no circumbinary disc is formed onany scale.

3.3 Prompt SNa explosions

Both the MBH and gas dynamics are una↵ected by feedbackfor the first 10 Myr as this is the assumed lifetime of massivestars (and hence for the onset of SNa feedback). To test

c 2015 RAS, MNRAS 000, 1–11

500 pc

x

y

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0

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-4 -3 -2log density

6 Pedro R. Capelo et al.

1 2 First pericentre 3 4 First apocentre

5 6 Second pericentre

End of the stochastic stage

7 Second apocentre 8 Third apocentre

9

End of the merger stage

10 11 12

End of the remnant stage

Figure 3. Stellar (red) and gas (blue) density snapshots (viewed face-on) at representative times of the 1:4 coplanar, prograde–progrademerger: (1) 0.20, (2) 0.30 (first pericentric passage), (3) 0.39, (4) 0.61 (first apocentric passage), (5) 0.88, (6) 0.97 (second pericentricpassage – end of the stochastic stage), (7) 1.05 (second apocentric passage), (8) 1.17 (third apocentric passage), (9) 1.24 (end of themerger stage), (10) 1.56, (11) 1.89, and (12) 2.21 Gyr (end of the remnant stage), respectively. We have run the simulations long enoughto capture the re-establishment of quiescence after the merger: note how the galaxy in the final snapshot is a normal-looking disc galaxy.The primary (secondary) galaxy starts the parabolic orbit on the left (right) of the first snapshot, moving right- (left-) wards. In orderto make the gas more visible, gas density was over-emphasized with respect to stellar density. Each image’s size is 70 × 70 kpc.

region has formed, with its radius oscillating between ∼60and ∼140 pc with the same temporal period of the BH ac-cretion. When the cavity reaches its maximum radius, theBH accretion is at its minimum, and vice versa. We believethat this is a clear case of BH self-regulation (‘breathing’),in which the BHs follow periodic stages of feeding and feed-back.

In the third panel of Figs 1 and 2, we show the SFrate (SFR) for three spherical regions centred around theBH (of radii 0.1, 1, and 10 kpc, respectively), and the totalSFR of the entire system. SFR is evaluated every 1 Myr,but here we show its average over the same time intervals asthose of gas mass and specific angular momentum, which areevaluated every 5 Myr. Central SFR (<100 pc) around theBH follows a similar behaviour to that of BH accretion rate,staying at low levels at all times except during the ∼300 Myrthat follow the second pericentric passage. During this time,central SFR around the secondary BH can increase by more

than three orders of magnitude from its previous levels andaccount for almost the totality of the SFR in the system.The increase in SFR around the primary BH is much moremodest, but in both cases it happens at the same time of theBH accretion rate increase. During the final stage, when thetwo BHs are at a mutual distance of !10 pc, central SFRis higher than during the first stage. Also, SFR around theprimary BH is more ‘centralized’: the SFR in the central kpccomprises most of the SFR of the inner 10 kpc, as opposed toduring the first stage. The link between BH accretion and SFis at the same time simple (both processes feed off the samereservoir of gas) and complex (the exact correlation betweenthem is still highly debated). In a separate paper (Volonteriet al. 2014, submitted), we present a detailed study on thistopic.

In the fourth panel, we show the amount of gas mass inthree spherical regions (of radii 0.1, 1, and 10 kpc, respec-

c⃝ 0000 RAS, MNRAS 000, 000–000

GALAXY MERGERS

on 100 kpc scales

SMBH Coalescence in Cosmological Galaxy Mergers 3

Figure 1. Group environment of the galaxy merger. The left panel shows a mock UVJ map of the galaxy group at z = 3.6. The whitecircle marks the virial radius of the group halo, while the green circles mark the merging galaxies. The upper-right and lower-right panelsshow a zoom-in on the central galaxy of the group and the interacting companion, respectively. Lengths are in physical coordinates.

Figure 2. From left to right: time evolution of the galaxy merger after the beginning of the re-sampled, higher-resolution simulation.Each panel shows a mock UVJ photometric image of the merger, and the red and blue dots mark the position of the primary and secondaryBH, respectively. Lengths are in physical coordinates.

to them. The orbital decay is governed by dynamicalfriction of the stellar cusps against the stellar, gas anddark matter background originating from the merger ofthe two hosts.

During the final stage of the merger (i.e. at t ≈ 20 Myrafter the particle splitting) the merger remnant is gaspoor (gas fraction ∼ 5%) owing to gas consumption bySF. Stars dominate the enclosed mass out to ∼ 3 kpcand provide the dominant contribution to the dynamicalfriction exerted by the background. Figure 4 shows themass distribution of the individual components when theseparation of the two SMBHs reaches about 300 pc, i.e.≈ 21.5 Myr after the particle splitting. The stellar mass

is almost 2 orders of magnitude larger than the gas oneover all spatial scales except in the central 10 pc, wherethe difference is about a factor of 20.

Then, we extract a spherical region of 5 kpc at t ∼21.5 Myr after particle splitting around the more massiveSMBH to initialize a direct N -body simulation contain-ing in total ∼ 6 × 106 particles. We treat the remaininggas particles in the selected volume as stars, since theyare sub-dominant in mass. Almost the entire stellar massis within 5 kpc, so an artificial cut-off at 5 kpc shall notintroduce significant changes in stellar mass profile in theinner region for follow up evolution. However, at trun-cation separation, the dark matter has a steeply rising

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Figure 5. Surface density maps, face-on and edge-on, for the control model (top panels) at 42 Myr and the clumpy model (bottompanels) at 345 Myr, with BH accretion, feedback, zero eccentricity and α = 10. Times are chosen to show the maximum separation ofthe secondary BH from the midplane in the first 400 Myr. In the control model, the secondary BH is always close to the midplane (themaximum distance in z-axis is much lower than ∼ 500 pc), because the clumps are small, disappear in the first ∼ 100 Myr and do notaffect BH dynamics. In the clumpy model, the secondary BH can exceed 1 kpc from the midplane and reach hotter and less dense regionsdue to clump interactions.

that the mass accretion rates for the central BH are gener-ally lower than in our case: indeed in their simulations theBH accretion rate is generally 10−5 M⊙ yr−1, whereas wefind ∼ 10−1 M⊙ yr−1 in the control case and ∼ 1 M⊙ yr−1 inthe clumpy case. The substantial difference is that they havepeaks that reach the Eddington rate, whereas we do not. Wecan compare our control and clumpy runs with standard BHaccretion and feedback to their models M16f10 and M4f50,respectively. This behaviour has likely two reasons: first ofall, in Gabor & Bournaud (2013), the aim was studying theBH growth in a clumpy medium, so they have only one cen-tral BH, into which all the produced clumps can accrete,while we have also the second BH that perturbs the mediumand can accrete clumps as well. The massive perturber isvery important in the control case, where gravity is not dom-inant like in the clumpy case, and indeed our galaxy has spi-ral arms also in the central part and a higher inflow, whereasthe model M16f10 of Gabor & Bournaud (2013) does not:the gas density map is quite smooth (see their Figure 1,bottom-left panel), except for some clumps in the outer

part, maybe due a different relaxation phase of the disc orto the short simulation time. Moreover, Gabor & Bournaud(2013) have BHs ten time less massive than in our mod-els (and therefore a lower Eddington mass accretion rate),which could explain the observed peaks at Eddington ratesin their mass accretion rates.

3.3 Limits of the isolated simulations; the“asymptotic” decay time-scale

In our initial conditions, the disc scale height is 0.05 Rd,where Rd ∼ 2 kpc. During the simulations, the vertical discextent remains below 1 kpc, even during the vigorous frag-mentation phase of the clumpy disc model (see Figure 5,edge-on views). This implies that, when the secondary BHis ejected out of the disc plane, the drag by dynamical fric-tion drops dramatically, and the orbital decay is correspond-ingly suppressed. As a result, a close BH pair never formsin most of the clumpy-disc runs. However, by constructionour galaxy models lack an extended dense spheroidal com-

MNRAS 000, 1–?? (0000)

Figure 5. Cartoon illustrating the journey travelled by SMBHs of masses in therange 106−8 M during major galaxy-galaxy collisions. The x− axis informs on theSMBH separation given in various panels, while the y− informs on the timescale. Thejourney starts when two galaxies (embedded in their DM halos) collide on kpc scales(right-most plot). The inset shows a selected group of galaxies from the cosmologicalsimulation described in Ref. [326]. The inset in the second panel (from right to left),from Ref. [327], depicts the merger of two disc galaxies and their embedded SMBHs.Pairing occurs when the two SMBHs are in the midst of the new galaxy that has formed,at separations of a few kpc. The SMBHs sink under the action of star-gas-dynamicalfriction. In this phase, SMBHs may find themselves embedded in star forming nucleardiscs, so that their dynamics can be altered by the presence of massive gas-clouds.Scattering off the clouds makes the SMBH orbit stochastic, potentially broadening thedistribution of sinking times during the pairing phase [328–331]. Furthermore, feedbackfrom supernovae and AGN triggering by one or both the SMBHs affect the dynamics asthese processes alter the thermodynamics of the gas and its density distribution, whichin turn affect the process of gas-dynamical friction on the massive BHs. It is expectedthat eventually the SMBHs form a Keplerian binary, on pc scales. Then, individualscattering off stars harden the binary down to the GW-driven domain. This is anefficient mechanism (and not a bottleneck) if the relic galaxy displays some degree oftriaxiality and/or rotation. In this case, there exists a large enough number of stars onlow-angular momentum orbits capable to interact with the binary and extract orbitalenergy [332]. The binary in this phase can also be surrounded by a (massive) circum-binary disc [333]. In a process reminiscent to type II planet migration, the two SMBHscan continue to decrease their separation and they eventually cross the GW boundary.Then, GW radiation controls the orbital decay down to coalescence.


common. However, the dynamical evolution is much faster in major mergers than inminor mergers: if the mass ratio of the merging halos and galaxies is too small, thesatellite halo is tidally disrupted early on in its dynamical evolution and its centralSMBH is left on a wide orbit, too far from the center of the larger galaxy to ever mergewith its SMBH [334, 335]. A population of wandering SMBHs is therefore predicted toexist in galaxies [336–338].

For mergers where the SMBHs can pair, i.e., find themselves in the newlyformed core of the merger remnant, the second hurdle is to get close enoughto form a gravitationally bound binary [339–342]. For SMBHs embedded in asingular isothermal sphere, a binary forms at a separation of a = GM/2σ2 '0.2 pc (MBBH/106 M) (σ/100 km s−1)−2, where the mass of the SMBHs exceeds theenclosed mass in stars, gas, and DM. The journey from the beginning of the merger,at tens of kpc, to the formation of the binary on pc to subpc scales, takes between afew tens of Myr for compact galaxies at very high redshift, z > 3− 4 to several Gyr forlarger, less dense galaxies at low redshift [339, 343, 344].

In the cases where the SMBHs form a bound binary within the Hubble time, thefinal crossing into the GW regime hinges on exchanges of energy through scatteringwith low angular momentum stars in the nucleus of the galaxy [345] or on extraction ofangular momentum through gravitational torques coming from a gas disc that may resulton the shrinking of its separation [346], or on a combination of the two processes. Recentresults of direct N-body simulations, Monte Carlo methods and scattering experimentsare giving an optimistic view of what has been considered to be the main bottleneckof the binary evolution for almost 40 years: the “final parsec problem,” i.e., runningout of low-angular momentum stars [345]. The evolution of SMBH binaries throughstellar scattering seems to continue at nearly a constant rate leading to merger in lessthan ∼1 Gyr [332, 347, 348] once rotation, triaxiality and the granularity of the stellardistribution are taken into account.

If the binary environment is dominated by gas, rather than stars, the binaryis expected to evolve through interaction with the accretion disk(s) surrounding theSMBHs, through the so called circumbinary disk phase. The disks are formed by thegas that inflows towards the SMBHs with an even small amount of angular momentum:in the single SMBH case these are referred to as “accretion discs” [349]. Depending onthe mass ratio q, the binary may or may not clear a gap within the circumbinary disk.In the former case (valid for q 1) the less massive hole behaves as a fluid elementwithin the accretion disk of the primary, thus experiencing what is known as Type Imigration. In the latter case (generally valid for q > 0.01), the formation of a centralcavity slows down the evolution of the binary (Type II migration). The energy andangular momentum transfer between the binary and the circumbinary disk is mediatedby streams leaking from the inner rim into the cavity. The pace of the supermassivebinary BH (SMBBH) evolution depends on the detailed dynamics of the streams crossingthe gap, which are partially accreted by the two SMBHs and partially flung back to thedisk, extracting the binary energy. Simulations generally concur that accretion into the


Figure 6. Relation between “timestart,” corresponding to the cosmic time of onset ofa galaxy-galaxy collision hosting SMBHs, and “ timecoal” at which the SMBHs mergefor circular binaries with the primary BH of 108 M and mass ratios q = 1, and 0.1.Redshifts at start and coalescence are given in the same plot. Dotted and dashed linesrefer to sinking times associated to the total of the merger time for the host halos andgalaxies, the pairing of SMBH, and then the shrinking of the binary subject to eitherstellar and gaseous processes. The black solid line represents “timecoal = timestart TheFigure shows that galaxies colliding at z ∼ 1 can display time delays as long as 3 Gyr.


central cavity is efficient enough to bring the SMBBH into the GW-dominated regime(∼ 0.01 pc) within ∼1-100 Myr [333, 350–356]. The role of AGN feedback on this scales,however, is still largely unexplored and may bring surprises.

In summary, SMBHs in merging galaxies have a long journey, that takes between1 and 10 Gyr depending on the cosmic time, mass and mass ratios of the hostgalaxies, galaxy morphologies, stellar and gas content, and on the masses of the SMBHthemselves. Figure 6 is an illustration of the delay timescales in the dynamics of SMBHmergers, during the pairing process in major mergers. This is a simplified example forcircular binaries with the primary SMBH of 108 M and mass ratios q = 1, and 0.1. Herewe included the time for halos to merge [357], and then added an additional timescalefor BHs to pair, based on the formalism of [358], motivated by the recent results of [335]who find that the halo merger timescale is insufficient to estimate the pairing timescale.We assumed 100 Myr from binary formation to coalescence for the gas-driven case andthe fit proposed by [348] for the stellar-driven case.

Theoretical studies and predictions, however, are still far from being complete: thisis a severely multi-scale problem, intimately connected with the processes of galaxyclustering on cosmological scales, which involves a rich physics. Determining thedistribution of merging times in SMBH mergers is critical, as only through the detailedknowledge of this distribution and associated processes, we are able to bracket theuncertainties in the estimates of the SMBH merger rates, relevant for the LISA mission,and to provide reliable estimate of the expected GW signal in the nHz band, relevantto PTAs.

8. Probing supermassive black hole binaries with pulsar timing arraysContributors: C. Mingarelli, M. Kramer, A. Sesana

Millisecond pulsars are excellent clocks over long timespans, making them idealtools to search for GWs [359–366]. Indeed, an array of millisecond pulsars forms agalactic-scale nanohertz GW detector called a Pulsar Timing Array (PTA, [367]). TheGWs change the proper distance between the Earth and the pulsars, which induces atiming delay or advance in the pulsar pulses. The difference between the expected pulsearrival time and the actual arrival time, called the timing residual, is used to search forsignatures of low-frequency GWs. The frequency band where PTAs operate is set bythe length of pulsar observations, and the cadence of the observations. Briefly, the lowerlimit is set by 1/Tobs, where Tobs is the total observation time, and the high-frequencylimit is set by 1/∆t, where ∆t is the cadence of the pulsar observations. This sets thesensitivity band between 1 nHz and 100 nHz.

The most promising signals in the PTA frequencies are due to the expected cosmicpopulation of SMBBHs. The systems of interest here have masses > 108 M, and canspend tens of millions of years emitting GWs in the relevant frequency band. Theincoherent superposition of their signals gives rise to a stochastic GW background(GWB, see, e.g. [368–372]). On top of it, particularly nearby and/or very massive


SMBBHs might be individually resolved [373, 374], providing interesting targets formultimessenger observations. Moreover, the memory effect following the merger of thosesystems may also give rise to detectable bursts of GW radiation [375, 376]. Gravitationalradiation may also originate from cosmic strings [377–379], and primordial GWs frome.g. inflation [380].

Here we focus on the stochastic GWB and continuous GW sources, but it is worthmentioning that the correlation function present in GWB searches depends on both theunderlying theory of gravity, the distribution of GW power on the sky, and the intra-pulsar distances. Indeed, additional GWB polarizations such as breathing modes canin principal be detected with PTA experiments, e.g. [381, 382], as can departures fromGWB isotropy [383–388]. Clustering of large-scale structure, resulting in an overdensityof merging SMBBHs, can lead to GWB anisotropy, as can nearby continuous GW sourceswhich are individually unresolvable, but can contribute to GWB anisotropy at level of∼ 20% of the isotropic component, see [374]. Moreover, pulsars separated by less than∼ 3 violate the short-wavelength approximation ms14, mm18, used to write down theHellings and Downs curve, and exhibit an enhanced response to GWs which may helpin their detection.

World-wide, PTA experiments have been taking data for over a decade, with effortsin North American governed by the North American Nanohertz Observatory for GWs(NANOGrav, see e.g. [389]), in Europe by the European PTA (EPTA) [390], andin Australia by the Parkes PTA (PPTA) [391]. The union of these PTAs forms theInternational Pulsar Timing Array (IPTA) [392], where the PTAs share data in theeffort to accelerate the first detection of nHz gravitational radiation, and to learn moreabout the individual millisecond pulsars. For a more comprehensive overview of PTAexperiments, see e.g. [364–366]

8.1. The Gravitational-Wave Background

The cosmic merger history of SMBBHs is expected to form a low-frequency GWB,which may be detected by PTAs in the next few years [393–395]. The time to detectiondepends strongly on the number of pulsars in the array, the total length of the dataset,and the underlying astrophysics which affects the SMBBH mergers, such as stellarhardening process, interactions with accretion disks, binary eccentricity, and potentiallySMBH stalling. Searches for the GWB have resulted in evermore stringent limits on theamplitude A of the GWB reported at a reference frequency of 1/yr:

hc = A

(f

yr−1

)−2/3

, (6)

where hc is the characteristic strain of the GWB [396]. This simple power-law scalingassumes the the SMBBH systems are circular when emitting GWs, and that theyare fully decoupled from their environment. Binary eccentricity and interactions withgas and stars around the binary can deplete the GW signal at very low frequencies(equivalently at wide binary separations), causing the GW strain spectrum to turn


ground-basedspace-basedgalaxy-based

aLIGO

Figure 7. The spectrum of gravitational radiation from low-frequency (PTA) to high-frequency (LIGO). At very low frequencies pulsar timing arrays can detect both theGWB from supermassive black hole binaries, in the 108 − 1010 M range, as well asradiation from individual binary sources which are sufficiently strong. We assume 20pulsars with 100 ns timing precision with a 15 year dataset for IPTA, and 100 pulsarstimed for 20 years with 30 ns timing precision for SKA. Both estimates assume 14-dayobservation cadence. This Figure was reproduced with minor changes from [415], whoin turn also used free software from [416].

over [397–400]. On the other hand, the amplitude of the GWB is affected by theabundance and mass range of the cosmic population of SMBHs. Therefore, futuredetections of a stochastic GWB will allow to constrain both the overall population ofSMBBHs and the physics driving their local dynamics [401–407]. Current non-detectionshave been used in [408] to challenge the popular MBH −Mbulge from [409]. However,a full analysis taking into account uncertainties in the merger rate, SMBBH dynamicsand the possibility of stalling is needed to draw firm conclusions [410].

The current upper limit on A from the various PTA experiments are similar andare improving with time. From the EPTA is A < 3× 10−15 [411], from the PPTA thisis A < 1 × 10−15 [391], from NANOGrav this is A < 1.45 × 10−15 [412], and the IPTAlimit is 1.7× 10−15 [392]. In order to improve those, further more sensitive observationsare needed, which can and will be provided by improved instrumentation and newtelescopes like FAST, MeerKAT and eventually the SKA [413]. Additionally, systematicsneeds to be addressed. For instance, solar system ephemeris errors can mimic a GWBsignal, if the underlying data are sufficiently sensitive [414]. Here, mitigation techniquescan be applied, as already done in the recent analysis of the NANOGrav 11-year data[412], while other effects, such as the interstellar weather may be best addressed withmulti-frequency observations. Future IPTA results will take those and other effects intoaccount.


8.2. Continuous Gravitational Waves

Individual nearby and very massive SMBBHs emitting nHz GWs can also bedetected with PTA experiments [417–420]. These SMBBHs are likely in giant ellipticalgalaxies, though the timescale between the galaxy merger and the subsequent SMBHmerger is poorly understood. Unlike LIGO or LISA sources, these SMBBHs are in thePTA band for many millions of years, and will likely merge outside both the PTA andLISA band. The sky and polarization averaged strain of a continuous GW source is

h =√

325M5/3

c [πf(1 + z)]2/3DL

, (7)

whereM5/3c = [q/(1 + q)2]M5/3 is the binary chirp mass, q < 1 is the binary mass ratio,

M is the total binary mass, f is the GW frequency and DL is the luminosity distanceto the binary.

The time to detection of continuous GW sources has been estimated in various ways:namely by simulating a GW source population based on cosmological simulations [373,394, 421], or by using an underlying galaxy catalog to estimate which nearby galaxies canhost SMBBH systems which are PTA targets [374, 422]. Both these approaches largelyagree that at least one SMBBH system will be detected in the next decade or so, withthe details depending on the amount of red noise in the pulsars. Detecting individualSMBBH systems is expected to shed light on the so-called final parsec problem, providingvaluable insights into how SMBBHs merge. In fact, pulsar distances are well-measured,it may be possible to measure the SMBH spins using the pulsar terms [423].

Importantly, resolving an individual SMBBH with PTAs will open new avenues inmultimessenger astronomy [424–427]. The combined GW and electromagnetic signalswill allow to pin down the properties of the binary and its environment, to study thedynamics of the SMBBH pairing and the coupling with the surrounding gaseous disc, ifpresent [364, 401]. Even a single joint detection will allow to nail down the distinctiveelectromagnetic properties of accreting SMBBHs. Analog systems can then be searchedin the archival data of large surveys, to quantify their overall cosmic population. Apromising way forward is to check candidates mined systematically in time domainsurveys for peculiar spectral signatures, hence producing credible targets for PTAs.

9. Numerical Simulations of Stellar-mass Compact Object MergersContributor: A. Perego

9.1. Motivations

Compact binary mergers (CBM) comprising at least one NS (i.e. BNS or BH-NSmergers) are unique cosmic laboratories for fundamental physics at the extreme. Thesepowerful stellar collisions involve all fundamental interactions in a highly dynamicaland intense regime (see, e.g., [428–431] and references therein for more comprehensiveoverviews). Their study is extremely challenging and requires multidimensional,


multiphysics and multiscale models. General relativistic hydrodynamical (GRHD)simulations, including a detailed, microphysical description of matter and radiation,are necessary to model the merger and to produce robust predictions for a large varietyof observables. Their comparison with observations will provide an unprecedented testfor our understanding of the fundamental laws of Nature, in regimes that will never beaccessible in terrestrial laboratories.

CBMs are events of primary relevance in astrophysics and fundamental physics.They are intense sources of neutrinos [432] and GWs [143], and primary targets for thepresent generation of ground-based GW detectors. Both theoretical and observationalarguments support CBMs as progenitors of sGRB [433, 434]. At the same time, they arethe places where the heaviest elements in the Universe (including Gold and Uranium)are synthesized and ejected into space, via the so-called r-process nucleosynthesis [435–437]. The radioactive decay of these freshly synthesized, neutron-rich elements in theejecta powers a peculiar EM transient, called kilonova (or macronova), hours to daysafter the merger [438, 439]. Despite happening far away from the merger remnant, theγ-ray emission, the r-process nucleosynthesis, and the kilonova transient are extremelysensitive to physical processes happening where the gravitational curvature is stronger.In particular, the equation of state (EOS) of NS matter is thought to play a central rolein all the emission processes, since it determines the compactness of the merging NSs,their tidal deformations, the lifetime and spin frequency of the remnant, the amountand the properties of the ejected mass.

On August, 17th 2017, the first detection of GWs from a CBM event compatiblewith a BNS merger marked the beginning of the multimessenger astronomy era [21, 22]and remarkably confirmed several years of intense and productive lines of research. TheGW signal, GW170817, provided the link to associate a sGRB, GRB170817A [23], andthe kilonova AT2017gfo transient [181, 440, 441] to a CBM localized in the NGC4993galaxy. Moreover, this single event set constraints on the EOS of NS matter and gavean independent measure of the Hubble constant [24].

Many relevant aspects of the merger and of the emission processes are, however,not yet fully understood and relate to open questions in fundamental physics andastrophysics. The interpretation of the observational data strongly relies on thetheoretical modeling of compact binary sources in GR. During the last years theunprecedented growth of computing power has allowed the development of increasinglysophisticated numerical models. The most advanced simulations in Numerical Relativity(NR) are set up using a first-principles and an ab-initio approach. Neutrino radiationand magnetic fields are thought to play a key role during the merger and its aftermath.Their inclusion in detailed NR simulations, in association with microphysical, finitetemperature EOS for the description of matter at and above nuclear saturation densityis one of the present challenges in the field. Reliable predictions do not only require theinclusion of all the relevant physics, but also accurate numerical schemes and numericallyconverging results. Robust and computationally stable discretizations are also essentialto perform long-term and high-resolution simulations.


9.2. Recent results for binary neutron star mergers

9.2.1. Gravitational waves and remnant properties A primary outcome of BNSsimulations are accurate gravitational waveforms. Numerical models provide consistentand continuous GW signals for all three major phases: inspiral, merger, and post-merger/ring-down. As discussed earlier, key parameters encoded in the waveform arethe masses and the spins of the coalescing objects, the quadrupolar tidal polarizability(depending on the NS EOS), and possibly the residual binary eccentricity. Forthe inspiral phase, state-of-the-art simulations cover at least 10 to 20 orbits beforecoalescence [442, 443]. Due to the cold character of this phase, zero-temperature EOSsfor the NS matter are often used. High-order finite difference operators have beenshown to be key to reach high-confidence numerical results [444]. Error control on thenumerical results is essential for using NR waveforms for the analysis of data streamscoming from GW detectors. State-of-art analysis of the error budget include the studyof the numerical convergence, requiring multiple resolutions (4 or 5), and an estimate ofthe error due to the finite distance extraction of the GW signal within the computationaldomain. Only recently, initial conditions for rotating NSs have become available [445].They were used to follow consistently, for the first time, the inspiral phase in numericalrelativity, including spin precession as well as spin-spin and spin-orbit couplings [446–448]. The high computational costs have limited the exploration of the BNS wideparameter space. However, thanks to the large set of presently available waveforms,detailed comparisons with Analytical Relativity results (including post-Newtonian andEffective One Body approaches) are nowadays possible [442, 449, 450]. Moreover, largedatabases of NR waveforms led to the production of purely NR-based waveform models[451]. These results are critical to improve the quality of semi-analytic waveformsemployed in current GW data analysis.

The merger and its aftermath are highly dynamical and non-linear phases. Asystematic study of the remnant properties and of its GW emission is presently madepossible by large and extensive sets of simulations in NR (e.g., [452–454]), sometimesextending for tens of ms after the merger. Detailed analysis of the post-mergergravitational waveforms revealed the presence of characteristic high-frequency peaks,which will be possibly accessible by third generation GW detectors. These features inthe GW spectrum are associated with properties of the nuclear EOS [452, 455, 456],with the presence of a magnetised long-lived massive NS [457], with the development ofone-arm spiral instabilities [458–460] or convective excitation of inertial modes inside theremnant [461]. However, a firm understanding of these structures in the GW frequencyspectrum is still in progress. The remnant fate depends primarily on the masses of thecolliding NS and on the nuclear EOS. The collapse timescales of metastable remnants toBHs crucially relies on physical processes that are still not fully understood, includingangular momentum redistribution (possibly of magnetic origin) and neutrino cooling.A prompt collapse to a BH, expected in the case of massive enough NSs and soft EOS,results in the most luminous GW emissions at merger. For stiffer EOS or lower NS


masses, differential rotation and thermal support can temporarily prevent the collapseof an object whose mass is larger than the mass of a maximally rotating NS, forming a so-called hypermassive NS (HMNS). Mergers producing a HMNS emit the largest amount ofenergy in GWs, since an intense luminosity is sustained for several dynamical timescales.If the remnant mass is below the maximum mass of a maximally rotating NS or even ofa non-rotating NS, a supramassive or massive NS can form, respectively. Gravitationalwaveforms for a supramassive or massive NS are similar to those of a HMNS case,but weaker. The detection of this part of the spectrum is beyond the capability ofthe present generation of GW detectors [462]. NR simulations take consistently intoaccount the angular momentum emitted in GWs and in ejected mass, and predict theangular momentum of the final remnant to be 0.6 . J/M2 . 0.85. In the HMNSand supramassive NS cases, the super-Keplerian value of J provides a large reservoir topower subsequent angular momentum ejections. Within the first dynamical timescale,this leads to the formation of a massive, thick disk around the central remnant.

9.2.2. Matter ejection and electromagnetic counterparts Numerical simulations arenecessary to quantitatively model matter ejection from BNS. Different ejectionmechanisms result in different ejecta properties. In addition to GW emission, BNSsimulations predict the ejection of both tidal (cold) and shocked (hot) dynamic ejecta,a few ms after the GW peak [452, 453, 463–467]. The most recent NR simulationsemploying microphysical, finite temperature EOS predict between 10−4 and a few 10−3

M of dynamic ejecta, moving at v ∼ 0.3 c. This range covers both intrinsic variability inthe binary (e.g., NS masses) and uncertainties in the nuclear EOS. However, results forthe same systems from different groups agree only within a factor of a few. Differencesin the treatments of the NS surface and of the floor atmosphere, as well as in themicrophysical content and in the extraction of matter properties at finite radii possiblyaccount for these discrepancies. Differently from what was previously thought, state-of-the-art simulations including weak reactions show that neutrino emission and absorptiondecrease the neutron richness of the equatorial ejecta and, more significantly, of the highlatitude ejecta. The former can still synthesize the heaviest r-process elements [468],while for the latter the formation of lanthanides can be inhibited. These results havebeen recently confirmed by parametric studies based on NR models [469, 470].

Long term (100s of ms) simulations of the merger aftermath show wind ejectacoming from the remnant and the accretion disk. These winds are powered by neutrinoabsorption [471, 472] or, on the longer disk lifetime, by viscous processes of magneticorigin and nuclear recombination inside the disk [465, 473–476]. If a HMNS has formed,the neutrino emission is expected to be more intense and the differential rotation canalso power winds of magnetic origin [477]. The unbound mass can sum up to several10−2 M, depending on the disk mass, and moves at ∼ 0.1 c. Weak processes, includingneutrino irradiation, decrease the neutron richness, particularly at high latitudes andfor very long-lived remnants [478]. This implies the production of the first r-processpeak in ν-driven winds and viscous ejecta. In the former case, the nucleosynthesis of


lanthanides is highly suppressed, while in the latter the wider distribution of neutronrichness leads to the production of possibly all r-process elements, from the first to thethird peak (e.g.,[474, 475, 478, 479]. Only a few simulations of the merger aftermath arepresently available and only few of them were performed in a NR framework. While thestudy of the viscosity-driven ejecta led to a partial agreement between different groups,simulations of ν- and magnetically-driven winds are still very few and additional workis required.

The different compositions and kinematic properties of the different ejecta channelshave implications for the magnitude, color, and duration of the kilonova (see [480]and [481] for recent reviews). The presence of lanthanides in the ejecta is expectedto significantly increase the photon opacity due to the presence of millions of mostlyunknowns absorption lines [482]. Detailed radiative transfer codes provide the mostaccurate models for the EM emission over timescales ranging from a few hours upto a few weeks [483–485]. Due to the uncertainties in the opacity treatment and inthe geometry of the ejecta, large differences could still arise between different models.More phenomenological approaches, characterized by significantly lower computationalcosts and accuracy, allow a more systematic and extensive exploration of the differentparameters in the ejecta properties [486, 487]. Based on these models, it was suggestedthat, due to the absorption of neutrinos at high latitudes, the presence of lanthanide-freematerial (characterized by lower opacity) could result in a bluer and earlier (∼ a fewhours) emission, compared with the redder and dimmer emission powered by materialenriched in lanthanides [471, 488, 489].

9.2.3. GW170817 and its counterparts The detection of GW170817 and of itscounterparts represents an unprecedented chance to test our understanding of BNS.The analysis of the gravitational waveform and the subsequent parameter estimationmade use of large sets of approximated waveform templates. The most recent resultsobtained in NR could potentially improve this analysis and set more stringent constraintson the intrinsic binary parameters and on the EOS of NS matter. The interpretation ofAT2017gfo as a kilonova transient revealed the need to consider at least two differentcomponents to explain both the early blue and the subsequent red emission [441, 490].More recently, the observed light curves have been traced back to the properties and tothe geometry of the ejecta predicted by the most recent numerical models. This analysishas confirmed the multi-component and anisotropic character of the BNS ejecta, aswell as the central role of weak interaction in setting the ejecta composition [491–493].Finally, the combination of the information extracted from the GW and from the EMsignal enabled the possibility to set more stringent constraints on the nuclear EOS, in agenuine multimessenger approach [494, 495]. Both very stiff and very soft EoSs seem tobe disfavored by this kind of analysis. A similar approach has been used to show thatthe formation of a HMNS is the most probable outcome of this event [496].


9.3. Recent results for black hole-neutron star mergers

Modelling of BH-NS mergers in GR shares many similarities with BNS mergermodelling, but reveals also differences and peculiarities (see [429] for a review). Thelarger ranges in masses and spins expected for stellar mass BHs significantly increasethe parameter space. The large mass ratio between the colliding objects makes NRsimulations more expensive, since the inspiral requires larger initial distances to covera sufficient number of orbits, and the dissimilar lengthscales require higher resolutions.These numerical challenges have limited the number of models and waveforms that arepresently available in comparison with the BNS case. If the final remnant is alwaysrepresented by a spinning BH, the presence of a disk around it depends on the massratio and BH spin [497–499]. Larger BHs, with moderate spins, swallow the NS duringthe plunge dynamical phase, while the decrease of the last stable orbit in the caseof aligned, fast spinning BHs leads more easily to the tidal disruption of the NS (inthe form of mass shedding) and to a massive accretion disk formation (with massespossibly in excess of 0.1 M). Misalignment between the orbital and the BH angularmomentum induces non-trivial precessions, encoded both in the GW signal and in theamount of mass outside the BH horizon. The GW spectrum shows a characteristiccutoff frequency directly related with the orbital frequency at the NS tidal disruptionor at the last stable orbit. Since this frequency decreases with increasing BH spin andwith decreasing compactness, its detection could provide direct constraints on the NSmatter EOS. Dynamical mass ejection from BH-NS merging system is expected to becaused by tidal torque when the NS is disrupted, and to happen predominantly alongthe equatorial plane. The ejected mass can be significantly larger than in the BNScase and neutrino irradiation is expected to have only a minor effect in changing theejecta composition [500]. Similarly to the BNS case, the viscous evolution of the diskdrives significant matter outflows on the disk lifetime, while the absence of a central NSreduces the emission of neutrinos and the ejection of ν- or magnetically-driven winds[501]. The weaker effect of neutrino processes on the dynamic ejecta results alwaysin robust r-process nucleosynthesis between the second and the third r-process peaks,while viscous disk winds can still synthesize all r-process nuclei. The high lanthanidecontent in the ejecta is expected to produce a redder kilonova transient a few days afterthe merger [502].

9.4. Perspectives and future developments

A complete and dense exploration of the wide parameter space, including aconsistent treatment of NS and BH spins and of their evolution, is one of the majorchallenges in the study of CBMs in NR. The extraction of waveforms from long inspiralsimulations at high resolution, employing accurate high-order numerical schemes, isnecessary to build a robust NR waveform database. These templates, in combinationwith more analytical approaches, can guide the construction of complete and coherentsemi-analytical waveforms for the GW data analysis, spanning many orbits from the


inspiral to the actual coalescence. The large variety of properties in the collidingsystem potentially translates to a broad distribution of properties for the remnantand for the ejecta. Long term simulations of the merger and of its aftermath arethe necessary tool to provide a complete and accurate description of the ejecta. Theinclusion of the physics necessary to model the remnant and the matter ejection isstill at its outset. In particular, the consistent inclusion of neutrino radiation andmagnetic field evolution in NR model is extremely challenging. Leakage scheme forneutrino radiation are presently implemented in many CBM models (see, e.g., theintroduction section of [503] for a detailed discussion). They provide a robust andphysically motivated treatment for neutrino cooling. However, they are too inaccurateto model long term evolution and neutrino absorption in optically thin conditions.State-of-the-art simulations include gray moment scheme [466, 504]. Nevertheless, thesignificant dependence of neutrino cross-sections on particle energy requires spectralschemes. Large velocity gradients inside the computational domain make the accuratetransformation of the neutrino energy spectrum between different observers and itstransport highly non-trivial. Moreover, the application of moment schemes for collidingrays in free streaming conditions leads to closure artifacts in the funnel above theremnant. Monte Carlo radiation schemes represent an appealing alternative, but theircomputational cost is still far beyond our present capabilities. The usage of Monte Carlotechniques to provide more physical closures in moment scheme seems a more viableapproach [505]. Neutrino masses have a potentially high impact on the propagation andflavor evolution of neutrinos from CBMs. New classes of neutrino oscillations, includingthe so-called matter neutrino resonances, can appear above the remnant of BNS andBH-NS mergers and influence the ejecta nucleosynthesis [506, 507]. However, numericalmodelling of these phenomena is still at its dawn [479, 508, 509].

General relativistic magneto-hydrodynamics codes are presently available and theyhave started to access the spatial scales necessary to consistently resolve the magneto-rotational instabilities (MRIs, [510]). The latter appear in the presence of differentialrotation and locally amplify the magnetic field. However, global simulations obtainedwith unprecedented spatial resolution (of the order of 12.5 m) do not show convergenceyet, proving that we are still not able to resolve the most relevant scales for magneticfield amplification in a self-consistent way [511]. In the past, sub-grid models have beenused as alternative approaches to ab-initio treatments [457, 512, 513]. More recently,two different formulations of effective MHD-turbulent viscosity in General Relativityhave been proposed and used in both global BNS and long term aftermath simulations[514, 515]. Moreover, the role of bulk viscosity in the post-merger phase has startedbeing investigated [516].

Accurate modelling of the GW and EM signals for CBMs are key to set tightconstraints on the EOS of NS matter, which still represents one of the major sourcesof uncertainty in both fundamental physics and in numerical models. If the connectionbetween CBMs and short GRBs will be confirmed by future detections, the knowledgeof the remnant fate, as well as of the environment around it, will be crucial to address


the problem of short GRB engines, including jet formation, collimation, and break-out.The accurate modelling of small-scale magnetic field amplification, as well as of heatredistribution due to neutrino transport, is key to predict the lifetime of the remnant forcases of astrophysical interest. In fact, the presence of highly rotating and magnetizedmassive NSs [457, 517], or of fast spinning BHs, is anticipated to play an essential rolein solving the puzzle of the short GRB central engine.

The high computational costs required by long-term, high-resolution, numericallyaccurate and multiphysics models of CBMs point to the need of developing anew generation of numerical schemes and codes for the new generations of largesupercomputers. These codes will need, for example, to improve scalability and toemploy more heavily vectorization in the hybrid (shared & distributed) parallelizationparadigm. This is perhaps the greatest challenge in the NR field for the years to come.

10. Electromagnetic Follow-up of Gravitational Wave Mergers

Contributor: A. Horesh

The year 2017 will be remembered as the year in which extraordinary achievementsin observational astrophysics have been made. On 2017, August 17, the LIGO andVirgo detectors detected for the first time GWs from the merger of two NSs, dubbedGW170817. Adding to the excitement was the detection of gamma-ray emission onlytwo seconds after the merger event, by the Fermi satellite. The sensational discoveryof a GW signal with a coincident EM emission led to one of the most comprehensiveobservational campaigns worldwide. A few hours after the GW detection, the LIGOand Virgo detectors managed to pinpoint the position of the GW event to an errorcircle of 34 sq. degrees in size (see Fig. 8 below). This area was small enough so thatan international teams of astronomers encompassing more than a hundred instrumentsaround the world on ground and in space could conduct an efficient search for EMcounterparts.

Roughly 11 hours after the GW event, an optical counterpart was announced bythe 1M2M (‘Swope’) project team [181, 518]. Many other teams around the worldindependently identified and observed the counterpart in parallel to the SWOPE teamdiscovery. The optical detection of the counterpart pinpointed the source to a preciselocation in the galaxy NGC4993 at a distance of 40Mpc, (which is also consistent withthe distance estimate from the GW measurements; see Sec. 2). Overall, the counterpart(for which we will use the official IAU name AT2017gfo) was detected across thespectrum with detections in the ultraviolet, optical, infrared (e.g., [181, 440, 441, 518–527]) and later on also in the X-ray (e.g., [528–530]) and in the radio (e.g., [531, 532]).

Below we briefly summarize the observational picture with respect to theoreticalpredictions.


10.1. The High-energy Counterpart

For many years it was hypothesized that short gamma-ray bursts (GRBs) that areby now routinely observed, are originating from NS mergers (e.g., [533]). The theoreticalmodel includes the launching of a relativistic jet during the short period of accretiononto the merger remnant. This energetic jet is believed to be responsible for the promptgamma-ray emission. In addition, the interaction of the jet with the interstellar mediumwill produce an afterglow emission in the optical, X-ray and also radio wavelengths.

The short GRB (T90 = 2.0±0.5 sec) that has been discovered about 1.7 second afterthe GW170817 merger event [534] was irregular compared to other short GRBs observedso far. Most notable is the low equivalent isotropic energy Eiso ≈ 5 × 1046 erg (in the10− 1000 kev band), which is ∼ 4 orders of magnitude lower than a typical short GRBenergy. In addition to the initial detection by Fermi-GBM, there were observationsmade by the Integral satellite. Integral also detected the short GRB with a flux of∼ 1.4× 10−7 erg cm−2 (at 75− 2000 kev [535]).

Following the initial detection and once the candidate counterpart had beenlocalized, X-ray observations were made within less than a day of the merger event. Theinitial X-ray observations by both the Neil Gehrels Swift Observatory and by the NuclearSpectroscopic Telescope Array (NuSTAR) resulted in null-detections. However, later on,on day ∼ 9 after the merger, the Chandra telescope detected X-ray emission from theAT2017gfo for the first time [528] with an initial isotropic luminosity of≈ 9×1038 erg s−1.In the following days, the X-ray luminosity appeared to continue to slowly rise [528, 529].Late-time observations (109days after the merger) by Chandra still show that the X-rayemission continues to slowly rise [536].

In order to reconcile the relatively faint prompt gamma-ray emission and the lateonset of the X-ray emission, it was suggested that AT2017gfo was a regular shortGRB but one that is observed slightly off the main axis by ∼ 10 degrees of thejet [528]. However, this interpretation has yet to be tested against observations inother wavelengths, such as the radio (see below).

10.2. The Optical and Infrared Counterpart

Over four decades ago, a prediction was made by [435, 537] that neutron-richmaterial can be tidally ejected in a NS merger. About a decade later numericalsimulations also showed that NS mergers will exhibit such mass ejections, where themass ejected is expected to be in the mass range 10−5 − 10−2 M with velocities of0.1 − 0.3 c (e.g., [538]). It was also predicted that heavy elements would form in theneutron rich ejected material via r-processes. As discussed in detail in the previousSection 9, additional material is also expected to be ejected by accretion disk winds andfrom the interface of the two merging stars (the latter material will be ejected mostlyin the polar direction). In general, each ejected mass component may have a differentneutron fraction leading to a different r-process element composition.

As the ejecta from the NS merger is radioactive, it can power transient emission, as


proposed in Ref. [539]. The initial prediction was that the emission will be supernova-like and will be blue and peaking on a one-day timescale. Later on, more detailedcalculations by [183] showed that the peak of the emission will be weaker at a levelof 1041 erg s−1. The next theoretical developments were made by [482, 483, 486] whofor the first time calculated the effect of the heavy r-process element opacities on theemission. They found that due to very high opacities, the emission is expected to beeven weaker, with its spectral peak now in the infrared (instead of the optical) and withthe peak timescale being delayed. Since the various ejecta mass components may havedifferent compositions, they therefore may have different opacities. Thus in principle,one component may form a blue short-lived emission (the so-called ‘blue component’)while another may emit in the infrared with week-long timescales (the so-called ‘redcomponent’).

The optical emission from the AT2017gfo peaked at a faint absolute magnitude ofMV ≈ −16 in one day and began to rapidly decline, while the infrared emisson hada somewhat fainter and later (at approximately 2 days) peak compared to the opticalemission. The overall observed brightness of the source in the optical and infrared bandsand its evolution over time have shown in general an astonishing agreement with thepredicted properties of such an event.

The multiple photometric and spectroscopic measurements sets obtained by thevarious groups paints the following picture: At early times, at about 0.5 days afterthe merger, the ejecta temperature was high at T ∼ 11, 000K (e.g. [523]). A daylater the spectral peak was at ≈ 6000Angstrom, and the temperature decreased toT ≈ 5000K [440]. From this point onwards the spectral peak quickly moved into theinfrared band. The combined optical and IR emission and its evolution were found to beconsistent with an energy source powered by the radioactive decay of r-process elements(e.g. [441, 519, 520]). Based on both the photometric and spectroscopic analysis anestimate of the ejecta mass and its velocity were found to be Mej ≈ 0.02−0.05M witha velocity in the range 0.1− 0.3 c [440, 519–523, 525–527].

One of the main claims with regards to this event is that the observations provideevidence for the formation of r-process elements. This conclusion is mainly drivenbased on the evolution of the infrared emission [519, 524]. [441], for example, obtainedlate-time IR measurements using the Hubble Space Telescope and argue that the slowerevolution of the IR emission compared to the optical require high opacity r-process heavyelements with atomic number > 195. The infrared spectrum shows broad features whichare presumably comprised of blended r-process elements [441]. These broad spectralfeatures were compared to existing model predictions and show a general agreement,albeit there are still inconsistencies that need to be explained (e.g., [481, 524, 540]).There is still an ongoing debate about the composition of the heavier elements. Ref. [520]claims that the observed infrared spectral features can be matched with CS and Telines. Others estimate the overall fraction of the lanthanides in the ejecta and findit to be in the range Xlan ≈ 10−4 − 10−2. It seems that at early times (up to 3-5days), the ejecta that dominates the emission has very low lanthanide fraction and that


there are discrepancies between the predicted and observed optical light curves (e.g.,[522, 526]. Several works (e.g., [440, 481, 520, 521, 524, 527, 540, 541]) explain the earlyvs. late-time behaviour of the emission by having an ejecta with two components (asalso predicted in the literature). The first component is the “blue” kilonova, with ahigh electron fraction and thus a low fraction of heavy elements, which dominates theoptical emission early only. A “red” kilonova is the second component that is comprisedby heavy r-process elements and that produces the slow IR evolution and the broadspectral features observed at late-times. Still there are some claims that even at latetimes, the emission can be originating from an ejecta with only low-mass elements [542].

10.3. The Radio Counterpart

In addition to the radio afterglow emission on the short time scale (days), radioemission is also expected on longer time scales (month to years). The latter is not aresult of the relativistic jet but rather originating from the interaction of the slowerdynamical ejecta with the instersteIlar medium (ISM) [543]. This emission is expectedto be rather weak where the strength of the peak emission depends on the velocity ofthe dynamical ejecta, the ISM density and on some microphysical parameters. Whilein the past, radio afterglows of short GRBs (not accompanied by GWs) were detected(e.g., [544]), long-term radio emission was never observed until GW170817 (includingin the previous cases where kilonova candidates were discovered [545, 546]).

Similar to any other wavelength, radio observations were undertaken within the dayof the GW170817 merger discovery and the following days. The early time observationsperformed by The Jansky Very Large Array (VLA), The Australian Compact Array(ATCA), the Giant Meterwave Radio Telescope (GMRT) and the Atacama LargeMillimeter Array (ALMA), all resulted in null-detections [531, 547, 548]. Late-timeVLA observations, however, finally revealed a radio counterpart ≈ 16days after themerger [531]. The initial radio emission was weak at a level of a couple of tens ofµJy only (at both 3 and 6GHz). Follow-up ATCA observations confirmed the radiodetection. Upon detection, a long-term radio monitoring campaign was initiated, andthe results are reported in Ref. [532]. They report that the radio emission still continuedto rise at > 100days after the merger. The multiple frequency observations also showthat the radio emission is optically thin with a spectral index of α = −0.6.

[531] compared the radio observations to several predictions including an on-axisjet, a slightly off-axis jet, and being completely off-axis. In addition, they included intheir comparison a model in which the jet forms a hot wide-angled mildly relativisticcocoon. This cocoon formed as the jet is working its way out of the dynamical ejecta,may also lead to gamma-ray, X-ray, and radio emission, but with different characteristicsthan the emission formed by either the highly relativistic jet or the slower dynamicalejecta. In fact, [531] find that both an on-axis jet model, or even a slightly off-axisjet one, are expected to produce bright radio emission in the first few days after themerger which by day 16 should start fading, a prediction which does not match the rising


observed radio source. The model that is the most consistent with current data is thecocoon model with either a choked or successful or ‘structured’ jet (see e.g., [536, 549]).In fact, the choked-jet cocoon model may also explain the relatively low energy of thegamma ray emission [550].

If both the late-time radio and X-ray emission originate from interaction of thematerial (whether it is the dynamical ejecta or a cocoon) with the ISM, one can testwhether the observed emission in both wavelengths is indeed connected as expected.For now, it seems that the observed X-ray emission fits the prediction which is basedon extrapolating the observed radio emission into higher energies. Both the X-rays andthe radio emission have also roughly the same spectral slope. This suggests that thereis no additional power source in play at this time.

10.4. Many Open Questions

While vast amounts of data have been collected for this amazing merger event, andwhile a flood of papers report many types of analysis and conclusions (by no means arewe attempting to cover all of them here), there are still many open questions remaining.For example, is this event connected to short GRBs or do we still have to prove thisconnection? Which r-process elements form and at what time? Do the observationsreally require producing heavy r-process elements or can they all be explained by lighter-element components? How many components does the ejecta have and which onedominates the emission and when? Are there any other emission power sources inplay such as a magnetar (even if at short time) ? Are we really seeing a cocoon witha choked or successful jet or is there some other scenario that can explain the radioemission combined with all the other evidence?

As scientists around the world are still working on analyzing all the data in handfor this event and are also still collecting new data, they are also gearing up for thefuture, and preparing for the next year when the LIGO and Virgo detectors switch backon. At this time, hopefully more events with EM signatures will be discovered and moreanswers (and surely more new questions) will present themselves.

11. X-ray and gamma-ray binariesContributor: M. Chernyakova

The population of Galactic X-ray sources above 2 keV is dominated by the X-raybinaries, see e.g. [551]. A typical X-ray binary contains either a NS or a BH accretingmaterial from a companion star. Due to angular momentum in the system, accretedmaterial does not flow directly onto the compact object, forming a differentially rotatingdisk around the BH known as an accretion disk . X-ray binaries can be further dividedinto two different classes, regardless the nature of the compact object, according to themass of the companion star: high-mass X-ray binaries and low-mass X-ray binaries. Thesecondary of low-mass X-ray binary systems is a low-mass star, which transfers matter


by Roche-lobe overflow. High-mass X-ray binaries comprise a compact object orbiting amassive OB class star. High-mass X-ray binaries systems are a strong X-ray emitter viathe accretion of matter from the OB companion. At the moment 114 high-mass X-raybinaries [552] and 187 low-mass X-ray binaries [553] are known.

Black hole X-ray binaries are interacting binary systems where X-rays are producedby material accreting from a secondary companion star onto a BH primary [349]. Whilesome material accretes onto the BH, a portion of this inward falling material may alsobe removed from the system via an outflow in the form of a relativistic plasma jet or anaccretion disk wind, see e.g. [554] for a review. Currently, the known Galactic BH X-raypopulation is made up of 19 dynamically confirmed BHs, and 60 BH candidates [555].The vast majority of these Galactic BH X-ray objects are low-mass X-ray binaries. Mostof these systems are transient, cycling between periods of quiescence and outburst. Thisbehaviour is associated with changing geometries of mass inflow and outflow, e.g. [554].

At higher energies, however the situation is drastically different. While currentCherenkov telescopes have detected around 80 Galactic sources (see the TeVCatcatalogue [556]), only 7 binary systems are regularly observed at TeV energies.Properties of PSR B1259-63, LS 5039, LSI +61 303, HESS J0632+057 and 1FGLJ1018.6-5856 are reviewed in [557]. Since 2013 two more Galactic binaries have beendiscovered at TeV sky, PSR J2032+4127 [558] and HESS J1832-093 [559], but still thenumber of binaries observed at TeV sky is extremely small, and the reason why thesesystems are able to accelerate particles so efficiently is not known yet. These systems arecalled gamma-ray-loud binaries (GRLB), as the peak of their spectral energy distributionlies at GeV - TeV energy range.

All GRLB systems host compact objects orbiting around massive young star of O orBe spectral type. This allows to suggest, that the observed γ-ray emission is producedin the result of interaction of the relativistic outflow from the compact object with thenon-relativistic wind and/or radiation field of the companion massive star. However,neither the nature of the compact object (BH or NS?) nor the geometry (isotropic oranisotropic?) of relativistic wind from the compact object are known in the most cases.Only in PSR B1259-63 and PSR J2032+4127 systems the compact object is known tobe a young rotation powered pulsar which produces relativistic pulsar wind. Interactionof the pulsar wind with the wind of the Be star leads to the huge GeV flare, duringwhich up to 80% of the spin-down luminosity is released [560, 561].

In all other cases the source of the high-energy activity of GRLBs is uncertain. Itcan be either accretion onto or dissipation of rotation energy of the compact object.In these systems the orbital period is much shorter than in PSR B1259-63 and PSRJ2032+4127, and the compact object spend most of the time in the dense wind of thecompanion star. The optical depth of the wind to free-free absorption is big enough tosuppress most of the radio emission within the orbit, including the pulsed signal of therotating NS [562], making impossible direct detection of the possible pulsar.

In Ref. [563] authors tried to deduce the nature of the compact source in LSI +61303 studying the relation between X-ray luminosity and the photon index of its X-ray


spectrum. It turned out that existing X-ray observations of the system follows the sameanti-correlation trend as BH X-ray binaries [564]. The hypothesis on microquasar natureof LSI +61 303 allowed to explain the observed radio morphology [565] and explains theobserved superorbital period as a beat frequency between the orbital and jet-precessionperiods [566]. At the same time it was shown that the model in which the compactsource is a pulsar allowed naturally explain the keV-TeV spectrum of LSI +61 303 [562].Authors argued, that the radio source has a complex, varying morphology, and the jetemission is unlikely to dominate the spectrum through the whole orbit. Within thismodel the superorbital period of the source is explained as timescale of the gradualbuild-up and decay of the disk of the Be star. This hypothesis is also supported bythe optical observations confirming the superorbital variability of the Be-star disk [567].A number of multi-wavelength campaigns are currently ongoing aiming to resolve thenature of these peculiar systems.

GeV observations revealed a few more binaries visible up to few GeV. Among themare Cyg X-1 [568, 569] and Cyg X-3 [570]. However, contrary to the GRLBs describedabove these systems are transients and seen only during the ares, or, in the case of Cyg X-1, during the hard state. In addition to this the peak of the spectral energy distributionof these systems happens at much lower energies than in the case of binaries visible atTeV energies.

However contrary to the GRLBs described above these systems are transients andseen only during the flares, or ,in the case of Cyg X-1, during the hard state. Inaddition to this the peak of the spectral energy distribution of these system happensat much lower energies than in the case of binaries visible at TeV energies. From theseobservations it seems that wind collision can accelerate particles more efficient that theaccretion, but more sensitive observations are needed to prove it and understand thereason. Hopefully CTA [571] observations will be able to shed light on the details of thephysical processes taking place in these systems.

12. Supermassive black hole binaries in the cores of AGNContributors: E. Bon, E. M. Rossi, A. Sesana, A. Stamerra

Following mergers it is expected that the galaxy cores should eventually end up closeto each other. In this process, the term dual SMBHs refers to the stage where the twoembedded SMBHs are still widely separated (gravitationally-bound to the surroundinggas and stars and not to one another), while SMBBHs denotes the evolutionary stagewhere they are gravitationally bounded in the close-orbiting system of SMBHs.

Bound SMBBHs on centi-pc scales are the most relevant to GW emission (andtherefore to this roadmap). In the approximation of circular orbits, these systems emitGWs at twice their orbital frequency, i.e. fGW = 2/Porb & 1 nHz. As we discuss in thenext Section, this is the frequency at which PTAs are most sensitive [389, 391, 392, 411],having a concrete chance to make a direct detection of these systems within the nextdecade [393–395]. The presence of a SMBBH with an orbital period of several years


introduces a natural timescale in the system. In fact, numerical simulations of SMBBHsembedded in circumbinary accretion disks display consistent periodic behaviors of thegas leaking into the cavity [333, 351, 354, 355, 572, 573]. This led to the notion thatSMBBHs might be detected via periodicity of their lightcurve. However, the diffusiontime of the gas within the mini-disks surrounding the two holes is generally longer thanthe binary period, and it is not at all clear that the periodic supply of gas through thecavity will in turn result in periodicities in the binary lightcurves or their spectra [424].Even though the cross-gap accretion rate is generally periodic, only light generated at theaccretion stream/minidisk or outgoing stream/circumbinary disk shocks is guaranteedto follow this modulation [573, 574]. Since some periodicity seems inevitable, wefocus on this signature in the following discussion. We notice however, that severalspectral signatures of close SMBBHs have also been proposed, including a dimming atUV wavelengths [425], double Kα lines [424], notches in the spectral continuum [574],steepening of the thermal spectrum compared to the standard thin disk model [575].

Among many mechanisms proposed to explain the emission variability of activegalactic nuclei (AGN) besides outflows, jet precession, disk precession, disk warping,spiral arms, flares, and other kinds of accretion disk instabilities, one of the mostintriguing possibilities involves the existence of a SMBBH system in their cores [576–580], and the tidal disruption event (see Ref. [581] and references therein).

The light variability emitted from AGN was tracked much before they wererecognized as active galaxies. There are light curves showing AGN variability of over100 years of observations, see for example [582, 583], with variability timescale of overdecades. In fact, many AGN show variability of different time scales depending ontime scales of processes that drive the variability, such as speeds at which variationspropagate, for example the speed of light c ∼ 3 · 105 km/s or the speed of soundvs ∼ (kT/m)1/2, where c > vorb vs, or the time scale of the orbital motion vorb ∼(GM/R)1/2. The shortest timescale corresponds to the light crossing timescale, on whichthe reverberation mapping campaigns are based on. Orbital timescales are longer [584],and are dominant in the case of SMBBH systems. Recently, possible connection of AGNvariability time scales and orbital radius is presented in [585], indicating that variabilitytime scales may not be random, and that they might correspond to the orbiting timescales.

To identify possible candidates, we search for periodic variations in their light andradial velocity curves. We expect that periodic variability should correspond to orbitalmotion exclusively, while the other processes could produce only quasi periodic signals.Unfortunately, AGN were identified only about 70 years ago, so observing records arelong for few decades only, which is of order of orbiting time scales, and therefore not longenough to trace many orbits in historical light curves, not to mention the radial velocitycurves [345, 586–589], which are harder to obtain because of their faintness, and evenshorter records. Therefore, for AGN it is very hard to prove that the signal is actuallyperiodic, especially if they are compared to the red noise like variability curves, that infact, AGN light curves are very similar to [590, 591]. Therefore, standard methods like


Fourier and Lomb-Scargle [592] may show peaks of high looking significance but thederived p-value may not be valid [588, 591, 593].

Keeping in mind the aforementioned difficulties, a number of AGN have beenproposed to display a significant periodical variability in their light curves [579, 594–599]. Notable examples include the blazar OJ287 (11.5 yr period, [594, 598], the quasarPG1302-102 (6 yr period, [579]), the blazar PG 1553+113 (2 yr period, [600]). Amongthose candidates, there are only few that could indicate periodic light and radial velocitycurves in the same time [587–589, 601], which therefore could be recognized as SMBBHcandidates, like NGC4151 with a 15.9 year periodicity [587], NGC5548 with a ≈15year periodicity [588, 589], and Akn120 with a ≈20 year periodicity [601]. We notethat simulating the emission from such systems is very complex [573, 575, 602–604],especially for the eccentric high-mass ratio systems [577, 605, 606].

Among AGN, the class of blazars is dominated by the emission from the jet dueto beaming effects caused by the small angle of sight. Blazars show high variability inall wavebands from radio to gamma-rays. High energy emission is likely originated insmall jet scales and therefore can be modulated by the orbital motion of the SMBHbinary system. The modulation can be funneled through variations on the accretionrate induced by the perturbation on the disk by the companion SMBH, as suggestedfor OJ287, or in helical paths induced by precession [607] as suggested to explainthe clear signature of a periodic modulation on the gamma-ray blazar PG 1553+113[600]. More complex interplay among the different components in the jet, emitting atdifferent wavelength is possible in the framework of a binary SMBH system ([608]). Wemention that quasi-periodicities in the jet emission can be induced by intrinsic oscillatorydisk instabilities that can mimic periodical behaviour. The continuous gamma-raymonitoring of blazars by the Fermi-LAT satellite is providing new possible candidatesshowing periodic or quasi-periodic emission (see e.g., [609], [610]). Similarly, a 14 yearperiodicity is found in the X-ray and optical variations of 3C 273, while in OJ 287,the optical variability may not always be consistent with radio. Even, a detection ofperiodic variations of spinning jet could indicate presence of SMBBH [611].

The time domain window has only been opened in the past decade with dedicatedsurveys such as CRTS, PTF and Pan-STARRS, and already produced several SMBBHcandidates. Many of them, however, have been already severely constrained byPTA upper limits on the stochastic GW background they would imply [612]. Thisconfirms our poor understanding of SMBBH appearance. More sophisticated numericalsimulations, including 3-D grids, radiative transport schemes, feedback from theaccreting sources, etc., are needed to better understand the emission properties andpeculiar signatures of SMBBHs. Under their guidance, future candidates should be thenproposed based on systematic cross check of variability coupled with peculiar spectralfeatures.


12.1. Modeling electromagnetic signatures of merging SMBBHs

Although the identification of compact SMBBHs might have an important impacton GW observations with PTAs in the near future, looking further ahead, LISA isexpected to detect tens to hundreds of coalescing SMBBHs throughout the Universe.Electromagnetic observations related to the merger of SMBBHs are important both forcosmology and the astrophysics of galactic nuclei. Pinning down the host galaxy ofthe merger will allow us to combine redshift and distance from gravitational waves toconstrain cosmological parameters ([613], see next Section) and to study the large scalegalactic environment of merging SMBBHs, adding to our understanding to the processgalaxy formation. On the other hand, electromagnetic observations will give us accessto the properties of matter in the relative close environment of a merger and to the gasan stars (hydro)-dynamics as they adjust in response to the merger.

Given the importance of such identification, there has been an extensive effortto predict observable electromagnetic signatures that can occur in nearly coincidencewith the event (“prompt signals”) or afterwards (“afterglows”). In the following fewexamples will be given; notably, some of them also inspired recent models for possibleelectromagnetic counterparts to SOBBH mergers, of the kind detected by LIGO andVIRGO. As mentioned previously, SMBBH mergers, especially at high redshifts, canhappen in a gaseous environment that provides each SMBH with a “minidisc,” fedthrough streams leaking from a circumbinary disc. Those minidiscs are likely to beretained even after the orbital decay due to gravitational radiation dominates, providingdistinctive modulation of the emerging luminosity as the binary spirals in [614, 615].During the final orbits, the surviving gas between the black holes gets squeezed possiblyproducing super-Eddington outflows as discussed in [616, 617], but see [618]. Full GRsimulations have also been employed to study the possible formation of precessing jetsduring the inspiral and merger, in the attempt of identifying distinctive signatures [619–621]. General relativity predicts that a newly formed black hole suffers a recoil becauseGWs carry away a non-zero linear momentum (e.g., [622, 623]). This recoil affects thecircumbinary disc, bound to the black hole: a kick is imparted that shocks the gasproducing a slowly rising, ∼ 10 yr lasting afterglows [624–628]. In stellar mass blackhole merger, similar phenomena but on much shorter timescales can occur [172, 629].Contrary to the SMBH case, however, providing a gas rich environment for the mergeris a challenge. A possible venue involves cold relic discs, formed as a result of weaksupernovae, where accretion is suppressed until either the “squeezing” or the “kicking”heat them up again in the same configuration envisaged for SMBHs.

As already mentioned, the next theoretical challenge for these dynamical modelsis to predict realistic lightcurves and spectra, which will require non-trivial radiativetransfer calculations (see, e.g., [630]). With solid predictions in hand, appropriatestrategies can be devised to coordinate electromagnetic follow-ups, to take full advantageof multimessenger astronomy in the LISA era.


13. Cosmology and cosmography with gravitational wavesContributors: C. Caprini, G. Nardini, N. Tamanini

The recent direct measurement of GWs by the Earth-based interferometers LIGOand Virgo opened up a new observational window onto the universe and, right from thefirst detection, led to the discovery of a new, unexpected source: fairly massive stellar-origin BBHs. This demonstrates the great potential of GW observations to improve ourknowledge of the universe. Concerning cosmology, it is beyond doubt that the possibledetection of a stochastic GW background (SGWB) from the early universe would berevolutionary from this point of view: similar to the discovery of the CMB, whichconstitutes a milestone in our understanding of the universe, rich of consequences thatwe are still investigating. Furthermore, the plethora of new GW detections expected inthe next decades by both Earth and space-based interferometers will not only deliverfundamental information on the emitting astrophysical sources, but it will also bringcomplementary and independent data, with respect to standard EM observations, thatcan be used for cosmological purposes. In particular, by means of GW detections, wecan probe the history of the universe both at early and late times, shedding new lighton some of the most elusive cosmological mysteries, such as dark energy, DM and theorigin of cosmic inhomogeneities. In this Section, we overview how the observation ofGWs can enhance our knowledge of the history of the universe.

13.1. Standard sirens as a probe of the late universe

Within the theory of GR, a binary system composed by two compact astrophysicalobjects orbiting around each other, emits a GW signal with the two polarizations [631]

h×(t) = 4dL(z)

(GMc(z)

c2

)5/3 (πf(t)c

)2/3

sin[Φ(t)] cos ι ,

(8)

h+(t) = h×(t)1 + cos2 ι

2 cos ι cot[Φ(t)] , (9)

where h×,+(t) are the GW strains in the transverse-traceless gauge (we neglect herepost-Newtonian contributions). In these expressions ι is the orientation of the orbitalplane with respect to the detector, z is the redshift of the source, dL(z) is the luminositydistance, f is the GW frequency at the observer, Φ(t) is the phase of the GW, andMc(z) = (1 + z)(m1m2)3/5/(m1 + m2)1/5 is the redshifted chirp mass, with m1,2

being the masses of the two binary bodies. An accurate detection of the GW signalallows to reconstruct all the parameters in Eqs. (8) and (9) within some (correlated)uncertainties [632]. In particular, thanks to the reconstruction of dL, binaries can beemployed as reliable cosmological distance indicators.

Eqs. (8) and (9) highlight three key aspects of using inspiralling binaries ascosmological distance indicators: i) The measurement of dL from GW signals is notaffected by any systematic uncertainties in the modelling of the source, since the


dynamics of compact binary systems is directly determined by GR. This is in contrastwith supernovae type-Ia (SNIa), which require the cosmic distance ladder, i.e. crosscalibration with local measurements of sources of known distance, to overcome unknownsystematics in the determination of their luminosity distance. ii) Due to the scaling∝ d−1

L , GW cosmological indicators are suitable even at large distances where EMsources, whose intensity scales as d−2

L , are too faint. This implies that given the sameamount of emitted energy, a source producing GWs can be observed at higher distanceswith respect to a source emitting EM waves. iii) The measurement of the quotedwaveform does not allow to determine the redshift of the source: in fact Eqs. (9) and (8)are invariant under the transformation mi → mi(1 + z) plus dL → dL(1 + z). In otherwords the waveform detected from any system with masses mi at a distance dL will beequivalent to a waveform produced by a system with masses mi(1 + z) at a distancedL(1 + z).

The luminosity distance dL is tightly linked to the redshift z in a given cosmologicalsetup. For a homogeneous and isotropic universe (at large scale), the luminosity distanceis given by

dL(z) = c

H0

1 + z√Ωk

sinh[√

Ωk

∫ z

0

H0

H(z′)dz′], (10)

with Ωk being the present value of the density parameter of the spatial curvature, H(z)being the Hubble rate as a function of the redshift, and H0 = H(z = 0). Therefore,if besides dL (reconstructed from the waveform) one can independently establish theredshift z of a GW source, then one obtains a data point useful to constrain therelation (10). The parameters of any given cosmological model, which are implicitlyincluded in H(z) since its dynamics is determined by the Einstein equations, can thenbe statistically constrained by fitting Eq. (10) against a number of (z, dL(z)) data points.This can be done using observations from GW inspiralling binaries, if a determinationof their redshift is available by some means. For this reason, well detectable binaries,for which the redshift is also known (or at least estimated), are dubbed standard sirens,in analogy with SNIa used as standard candles in cosmography [633].

Of course, to obtain a robust bound, it is crucial to fit Eq. (10) with as manystandard sirens as possible at the most diverse redshifts, especially if the cosmologicalmodel at hand contains many parameters. The outcome of such an analysis can beremarkable. It constitutes the first robust cosmological test not using EM radiation asthe only messenger of astronomical information, and it allows to probe the validity ofthe cosmic distance ladder up to far distances. As demonstrated by the recent analysisof the LVC detection GW170817, discussed in section 13.1.2 below, it also providesa measurement of H0 that is independent of the calibrations necessary to establishconstraints using SNIa. In particular, if the current tension between theH0 measurementfrom SNIa and CMB (assuming ΛCDM) will persist, standard sirens will be a very usefulobservable to decipher the origin of this tension.

13.1.1. Redshift information


The cosmography procedure just described assumes that the redshift of the GW sourcescan be acquired. This is however not straightforward. Because of their intrinsicnature, coalescing BHs do not guarantee any signature besides GWs. Nevertheless,EM counterparts can be envisaged for BBHs surrounded by matter, or for compactbinaries where at least one of the two bodies is not a BH. For this reason, an EMcounterpart is expected for merging massive BHs at the centre of galaxies, which maybe surrounded by an accretion disk, and for BNSs, whose merger produces distinctiveEM emissions, including gamma ray bursts and kilonovae. On the other hand, extrememass ratio inspiral (EMRI) systems and stellar-origin BBHs (SOBBHs) are not expectedto produce significant EM radiation at merger, although their merging environment isstill unclear. Of course, in order to fit Eq. (10) with as many data points (z, dL) aspossible, it would help substantially to determine the redshift of all detectable standardsirens, independently of whether they do or do not exhibit an EM counterpart. There aremainly two different ways to obtain redshift information for a standard siren, dependingon the observation or not of an EM counterpart:

Method with EM counterpart: This method relies on EM telescopes to recognize thegalaxy hosting the GW source [632]. Reaching a good sky localization (O(10 deg2)or below) as soon as possible after the detection of the GW signal, is essential to alertEM telescopes and point them towards the solid angle determining the direction ofthe GW event to look for EM transients. Once such a transient is detected, the GWevent can be associated with the nearest galaxy whose redshift can be measuredeither spectroscopically or photometrically. It is important for this method to haveGW detectors able to rapidly reach a well-beamed sky localization of the GWsource. A network of GW interferometers not only improves the sensitivity to astandard siren signal (improving the reconstruction of dL) but also the identificationof the sky solid angle containing the source thanks to spatial triangulation.

Method without EM counterpart: This method allows to determine the sky localizationof the standard siren much after the GW event, with clear practical advantages,in particular, the presence of the source does not need to be recognized in realtime, and moreover the SNR required for a good sky localization does not needto be reached before the stage at which the EM signal might be triggered). Itadopts a statistical approach [632, 634]. Indeed, given a galaxy catalogue, thegalaxy hosting the standard siren has to be one of those contained in the box givenby the identified solid angle (with its experimental error) times a properly-guessedredshift range. This range is obtained by applying a reasonable prior to the redshiftobtained inverting Eq. (10). In this procedure one must of course take into accountthe dependency upon the cosmological parameters of H(z), which will affect thefinal posterior on the parameters themselves. The redshift of the standard sirencan be estimated as the weighted average of the redshifts of all the galaxies withinthe error box (an additional prior on the conformation of each galaxy may be alsoincluded). Due to the large uncertainty of this statistical approach, this method is


effective only when a limited amount of galaxies can be identified within the volumeerror box and only if a sufficiently large amount of GW events is observed.

These procedures suffer from some major uncertainties. The redshift appearingin Eq. (10) is the one due to the Hubble flow, and consequently the contributiondue to the peculiar velocity of the host galaxy to the measured redshift should besubtracted. Moreover, the real geodesic followed by the GW is not the one resultingfrom the homogeneous and isotropic metric assumed in Eq. (10), and in fact the lensingcontribution to dL due to cosmic inhomogeneities should be removed. Although inprinciple very precise lensing maps and galaxy catalogues are helpful to estimate suchsource of uncertainty [635–638], still the lensing and peculiar velocity effects have to betreated as a (large) systematic error that can be reduced only by means of numerousdetections (the uncertainty due to peculiar velocities dominating at low redshift, andthe lensing uncertainty dominating at large redshift).

13.1.2. Standard sirens with current GW data: GW170817The GW170817 event, corresponding to the coalescence of a BNS, is the exquisiteprogenitor of cosmography via standard sirens with EM counterpart. For that event, theLVC interferometer network recovered the luminosity distance dL = 43.8+2.9

−6.9 Mpc at 68%C.L., and a sky localization of 31 deg2 [21, 22], corresponding to the solid angle shown inthe left panel of Fig. 8 (green area). The optical telescopes exploring this portion of thesky identified an EM transient in association to the galaxy NGC4993, which is knownto be departing from us at the speed of 3327±72 km s−1. Although part of the peculiarvelocity of the galaxy NGC4993 could be subtracted [22], remaining uncertainties on thepeculiar motion eventually yield the estimate vH = 3017±166 km s−1 for its Hubble flowvelocity, corresponding to z ' 10−2 (notice that at this redshift the lensing uncertaintyis negligible). Using the Hubble law dL = z/H0, which is the leading order term in theexpansion of Eq. (10) at small z, one obtains the posterior distribution for H0 presentedin Fig. 8 (right panel), providing the measurement H0 = 70.0+12.0

−8.0 km s−1Mpc−1 at 68%C.L. [24]. The uncertainty on H0 is too large to make the measurement competitive withCMB [639] and SNIa constraints [640]. Nevertheless, this represents a local estimateof H0 that is not dependent on the cosmic distance ladder and the first cosmologicalmeasurement not relying only on EM radiation. Moreover, the large number of eventssimilar to GW170817 expected to be observed in the future will eventually yieldconstraints on H0 at the level of both local and CMB measurements.

13.1.3. Cosmological forecasts with standard sirensThe potential of current and forthcoming GW detectors for cosmology have been widelystudied in the literature. To appreciate the capability of standard sirens to probe thelate universe, here we report on the most recent forecasts (see e.g. Refs. [641–644] (LVC)and [645–650] (LISA) for previous studies). Such analyses are expected to continuouslyimprove over the upcoming years of data taking.


Figure 8. Left panel: The multi-messanger sky localization of GW170817 and theidentification of the host galaxy. Right panel: The posterior distribution of H0 comparedto the recent CMB and SNIa constraints [639, 640]. Figures taken from Refs. [22, 24].

Forecasts with ground-based interferometers: Ground-based interferometers can detectbinary systems composed of NSs and/or stellar-origin BHs, with masses rangingfrom few solar masses up to tens of solar masses. By means of the plannedLVC-KAGRA-LIGO India network, BNSs with counterpart will allow to puta tight constraint on H0. For example, assuming the ΛCDM model, H0 isexpected to be measured roughly with a ∼1% error after ∼100 detections withcounterpart [651, 652]. The result would be even tighter if, for some SOBBHs,the host galaxy could be identified, or if BNSs with counterpart are consideredinstead, as only ∼10 detections would yield the same level of accuracy [651, 653].The forecasts drastically improve for third generation ground-based detectors, suchas the Einstein Telescope (ET) [654–656]. In addition, besides the two approachesmentioned above for the redshift measurement, a further method, feasible onlywith an ET-like device, will become available [657] (see also [658] for a similaridea). Thanks to the tidal effects of a BNS waveform, the redshift-mass degeneracycan be broken. Consequently, by measuring the tidal effect in the waveform andassuming a prior knowledge on the NS equation of state, the redshift z entering theredshifted chirp mass can be reconstructed from the waveform itself. In this way,with more than 1000 detected BNS events, the ET is expected to constrain theparameters H0 and Ωm of the ΛCDM model roughly with an uncertainty of ∼ 8%and ∼ 65%, respectively [659]. However, if the rate of BNS mergers in the universewill result to be in the higher limit of the currently allowed range, the ET will beable to see up to ∼107 BNS mergers, ameliorating these constraints by two ordersof magnitude.

Forecasts with space-based interferometers: Space-based GW interferometry will openthe low-frequency window (mHz to Hz) in the GW landscape, which iscomplementary to Earth-based detectors (Hz to kHz) and PTA experiments (nHz).The Laser Interferometer Space Antenna (LISA) is currently the only planned spacemission designed to detect GWs, as it has been selected by ESA [64]. Several new


GW astrophysical sources will be observed by LISA, including SOBBHs, EMRIsand massive BBHs from 104 to 107 solar masses. These sources can not only beconveniently employed as standard sirens, but they will be detected at differentredshift ranges, making LISA a unique cosmological probe, able to measure theexpansion rate of the universe from local (z ∼ 0.01) to very high (z ∼ 10) redshift.The current forecasts, produced taking into account only massive BBHs [613](for which an EM counterpart is expected) or SOBBHs [660, 661] (for which noEM counterpart is expected), estimate constraints on H0 down to a few percent.However, joining all possible GW sources that can be used as standard sirenswith LISA in the same analysis, should not only provide better results for H0,which will likely be constrained to the sub-percent level, but it will open up thepossibility to constrain other cosmological parameters. The massive BBH datapoints at high redshifts will, moreover, be useful to test alternative cosmologicalmodels, predicting deviations from the ΛCDM expansion history at relatively earlytimes [662, 663]. Finally, more advanced futuristic missions, such as DECIGO orBBO, which at the moment have only been proposed on paper, may be able toprobe the cosmological parameters, including the equation of state of dark energy,with ultra-high precision [664–667]. They might also be able to detect the effectof the expansion of the universe directly on the phase of the binary GW waveform[668, 669], although the contribution due to peculiar accelerations would complicatesuch a measurement [277, 670].

13.1.4. Future prospectsThe recent GW170817 event has triggered numerous studies about the use of standardsirens for cosmography, and the forthcoming experimental and theoretical developmentsmight change the priorities in the field. As mentioned above, the network ofEarth-based GW interferometers is expected to detect an increasing number of GWsources employable as standard sirens, with or without EM counterparts. This willeventually yield a measurement of the Hubble constant competitive with CMB and SNIaconstraints, which might be used to alleviate the tension between these two datasets. Onthe other hand, higher redshift (z & 1) standard sirens data will probably be obtainedonly with third generation detectors or space-based interferometers. With these highredshift data we will start probing the expansion of the universe at large distances,implying that a clean, and not exclusively EM-based, measurement of other cosmologicalparameters will become a reality. Furthermore, the precision of cosmography viastandard sirens might be boosted by improvements in other astronomical observations.This is the case if e.g. galactic catalogues and lensing maps improve substantially in thefuture, as expected.

Concerning present forecasts, there is room for improvements, especially regardingthird generation Earth-based interferometers and space-borne GW detectors. The fullcosmological potential of future GW experiments such as ET and LISA has still to beassessed. For ET we still do not have a full cosmological analysis taking into account


all possible GW sources that will be used as standard sirens, using both the methodwith counterpart (BNSs) and the method without counterpart (SOBBHs and BNSs).Moreover, we still need to fully understand up to what extent the information on the NSequation of state can be used to infer the redshift of BNSs and NS-BH binaries, withthe latterpotentially representing a new interesting future source of standard sirens[642, 644, 671]. Regarding LISA, there are still several issues to be addressed in order toproduce reliable forecasts. For instance, the EMRI detection rate is still largely unknown[672], although these sources might turn out to be excellent standard siren candidatesat redshift 0.1 . z . 1 [673]. In addition, the prospects for massive BBH mergers asstandard sirens would be more robust if an up-to-date model of the EM counterpartwere implemented in the investigations. Performing these analyses and combining theresults for all the different types of standard sirens observable by LISA will allow for afull assessment of the cosmological potential of the mission.

Most of the literature on standard sirens assumes GR and an homogeneous andisotropic universe at large scales. However, breakthroughs may occur by generalizingthe aforementioned analyses to theories beyond GR, and in fact forecasts for scenariosnot fulfilling these assumptions are flourishing topics with potentially revolutionaryresults. For example, some models of modified gravity, formulated to provide cosmicacceleration, have been strongly constrained by the measurement of the speed of GW(compared to the speed of light) with GW170817 [674–677]. In general, in theoriesbeyond GR or admitting extra dimensions, the cosmological propagation of GWs changesand a comparison between the values of dL measured from GWs and inferred from EMobservations can provide a strong test of the validity of these theories (see [678–681] forrecent works).

Finally we mention that the large number of GW sources expected to be observedby third generation interferometers such as ET or by future space-borne detectors suchas DECIGO/BBO, might even be used to test the cosmological principle [682–684] andto extract cosmological information by cross-correlating their spatial distribution withgalaxy catalogues or lensing maps [685–687].

13.2. Interplay between GWs from binaries and from early-universe sources

The gravitational interaction is so weak that GWs propagate practicallyunperturbed along their path from the source to us. GWs produced in the early universethus can carry a unique imprint of the pre-CMB era, in which the universe was nottransparent to photons. In this sense, GW detection can provide for the first timedirect, clean access to epochs that are very hard to probe by any other observationalmeans. GWs of cosmological origin appear (pretty much as the CMB) to our detectorsas a SGWB [688, 689].

Several pre-CMB phenomena sourcing GWs might have occurred along thecosmological history, from inflation to the epoch of the QCD phase transition. Insuch a case, the SGWB would be constituted by the sum of all single contributions,


each of them potentially differing from the others in its spectral shape as a functionof frequency, or because of other properties such as chirality and/or gaussianity. It ishence crucial to have precise predictions of all proposed cosmological sources to possiblyisolate all components in the detected SGWB, with the aim of reconstructing the earlytime history of the universe.

On the other hand, binary systems can also be detected as a SGWB. This happensfor those (independent) astrophysical events that are overall too weak to be individuallyresolved. The level of their “contamination” to the cosmological SGWB thus dependson the sensitivity and on the resolution of the available detector. Crucially, at LIGO-like and LISA-like experiments the astrophysical component might be stronger thanthe cosmological one, so that the information hidden in the pre-CMB signal would beimpossible to recover, unless the astrophysical contribution is known in great detail andmethods are found to subtract it.

13.2.1. Status and future prospects

No measurement of the SGWB has been done yet, but upper bounds have been inferredfrom the observations. These are usually expressed in terms of the SGWB energydensity, which is given by ΩGW(f) = (f/ρc) ∂ρGW/∂f , with ρc and ρGW being thecritical and the SGWB energy densities, respectively. Specifically, by assuming thefrequency shape ΩGW(f) = Ωα(f/25Hz)α, the LVC found the 95% C.L. limit Ωα=0,2/3,3 <

17 × 10−7, 13 × 10−7 and 1.7 × 10−8 in the O(10) – O(100)Hz frequency band [690].The latest Pulsar Timing Array analysis, done with the data of the NANOGravcollaboration, yields h2Ωα=0 < 3.4×10−10 at 95% C.L. at 3.17×10−8 Hz [412]. Since theastrophysical sources are not expected to produce a SGWB with higher amplitude thanthese bounds, the latter are particularly relevant only for the most powerful cosmologicalsignals [380, 691].

However, it is probably only a matter of some years to achieve the first detection ofthe astrophysical SGWB by the LVC. Based on the current detection rates, the SOBBHsand BNSs lead to the power-law SGWBs ΩBBH(f) = 1.2+1.9

−0.9 × 10−9 (f/25 Hz)2/3 andΩBBH+BNS(f) = 1.8+2.7

−1.3 × 10−9 (f/25 Hz)2/3 in the frequency band of both LVC andLISA [692]. These signals are expected to be measurable by the LIGO and Virgodetectors in around 40 months of observation time [692], while in LISA they will reacha Signal to Noise Ratio (SNR) of O(10) in around one month of data taking [693]. Onthe other hand, given the large uncertainties on EMRIs [672], it is not clear whetherthey can also give rise to an observable SGWB component in the LISA band.

Focusing exclusively on the SGWB contribution from astrophysical binaries, twogeneral aspects about it are particularly worth investigating. The first regards thedetailed prediction and characterization of this component. For the time being,the literature does not suggest any technique to disentangle the cosmological SGWBcomponent from a generic astrophysical one. At present, it is believed that componentseparation will be (partially) feasible in the LISA data only for what concerns the


SGWB signal due to galactic binaries. In this case, indeed, the anisotropy of the spatialdistribution of the source (confined to the galactic plane), together with the motion ofthe LISA constellation, generates a SGWB signal modulated on a yearly basis, whichallows to distinguish and subtract this component from the total SGWB [694]. Althougha similar approach is not possible for the extra-galactic SGWB component, this examplewell illustrates that detailed predictions of the astrophysical signatures might highlightsubtraction techniques allowing to perform component separation and hopefully isolatethe information coming from the early universe (similarly to what done for foregroundsubtraction in CMB analyses).

The second aspect concerns the possibility of exploiting third generation detectors,such as the ET and Cosmic Explorer, to subtract the astrophysical SGWB component[695]. Their exquisite sensitivity may allow to resolve many of the SOBBHs and BNSsthat give rise to the SGWB in present detectors, including the full LVC-KAGRA-LIGOIndia network. This would clean the access to the cosmological SGWB down to a levelof ΩGW ' 10−13 after five years of observation [695]. This technique not only applies inthe frequency bandwidth of the Earth-based devices, but it can also be used to clean theLISA data and reach a potential SGWB of cosmological origin in the LISA band. Notethat LISA might as well help in beating down the level of the SGWB from SOBBHs byexploiting possible multi-band detections of the same source [696]. The SOBBH wouldbe detected first by LISA, during its inspiral phase; some years later, when the binaryhas arrived to the merger stage, it would reappear in Earth-based interferometers. Theselatter can therefore be alerted in advance, possibly leading to an increase in the numberof detected BBHs (depending on their LISA SNR).


Chapter II: Modelling black-hole sources of gravitationalwaves: prospects and challenges

Editor: Leor Barack

1. Introduction

The detection and characterization of gravitational-wave (GW) sources rely heavilyon accurate models of the expected waveforms. This is particularly true for blackhole binaries (BBHs) and other compact objects, for which accurate models are bothnecessary and hard to obtain. To appreciate the state of affairs, consider the followingthree examples. (i) While GW150914, the first BH merger event detected by LIGO, hadinitially been identified using a template-free search algorithm, some of the subsequentevents, which were not as bright, would likely have been altogether missed if template-based searches had not been performed. (ii) While the error bars placed on theextracted physical parameters of detected BH mergers have so far come primarily frominstrumental noise statistics, systematic errors from the finite accuracy of available signalmodels are only marginally smaller and would actually dominate the total error budgetfor sources of some other spin configurations or greater mass disparity; in the case of thebinary neutron star (BNS) GW170817, deficiencies in available models of tidal effectsalready restrict the quality of science extractable from the signal. (iii) Even with aperfectly accurate model at hand, analysis of GW170817 would not have been possiblewithin the timescale of weeks in which it was carried out, without the availability of asuitable reduced-order representation of the model, necessary to make such an analysiscomputationally manageable.

Indeed, accurate and computationally efficient models underpin all of GW dataanalysis. They do so now, and will increasingly do so even more in the future as moresensitive and broader-band instruments go online. The response of detectors like ET,and especially LISA, will be source-dominated, with some binary GW signals occurringat high signal-to-noise ratio (SNR) and remaining visible in band through many morewave cycles. The accuracy standard of models needs to increase commensurably withdetector sensitivity, or else modelling error would restrict our ability to fully exploit thedetected signals. As a stark example, consider that, in a scenario that is not unlikely,LISA’s output will be dominated by a bright massive BH (MBH) merger signal visiblewith SNR of several 100s. This signal would have to be carefully “cleaned out” ofthe data in order to enable the extraction and analysis of any other sources buriedunderneath; any model inaccuracies would form a systematic noise residual, potentiallyhiding dimmer sources. In the case of Extreme Mass Ratio Inspirals (EMRIs), whereO(105) wave cycles are expected in the LISA band at a low SNR, a precise model is acrucial prerequisite for both detection and parameter extraction.

This chapter reviews the current situation with regard to the modelling of GWsources within GR, identifying the major remaining challenges and drawing a roadmap


for future progress. To make our task manageable, we focus mainly on sources involvinga pair of BHs in a vacuum environment in GR, but, especially in Sec. 5, we also touchupon various extensions beyond vacuum, GR and the standard model (SM) of particlephysics.

Isolated vacuum BHs in GR are remarkably simple objects, described in exact formby the Kerr family of solutions to Einstein’s field equations (see, however, Sec. 8 ofthis chapter). But let two such BHs interact with each other, and the resulting systemdisplays a remarkably complicated dynamics, with no known exact solutions. Evennumerical solutions have for decades proven elusive, and despite much progress followingthe breakthrough of 2005 they remain computationally very expensive—prohibitivelyso for mass ratios smaller than ∼ 1 : 10—and problematic for certain astrophysicallyrelevant BH spin configurations. Systematic analytical approximations are possible andhave been developed based around expansions of the field equations in the weak-field orextreme mass-ratio regimes, and these may be combined with fully numerical solutionsto inform waveform models across broader areas of the parameter space. To facilitatethe fast production of such waveforms, suitable for GW search pipelines, effective andphenomenological models have been developed, which package together and interpolateresults from systematic numerical simulations and analytical approximations. Withthe rapid progress in GW experiments, there is now, more than ever, a need for aconcentrated community effort to improve existing models in fidelity, accuracy andparameter-space reach, as well as in computational efficiency.

The structure of this chapter is as follows. In Secs. 2 and 3 we review the twomain systematic approximations to the BBH problem: perturbation theory includingthe gravitational self-force (GSF) and post-Newtonian (PN) approaches, respectively.Section 4 surveys progress and prospects in the Numerical Relativity (NR) modelling ofinspiralling and merging BHs in astrophysical settings, and Sec. 5 similarly reviews therole of NR in studying the dynamics of compact objects in the context of alternativetheories of gravity and beyond the SM. Section 6 then reviews the Effective One Body(EOB) approach to the BBH problem, and the various phenomenological models thathave been developed to facilitate fast production of waveform templates. Section 7reviews the unique and highly involved challenge of data-analysis in GW astronomy,with particular emphasis on the role of source models; this data-analysis challengesets the requirements and accuracy standards for such models. Finally, Sec. 8 givesa mathematical relativist’s point of view, commenting on a variety of (often overlooked)foundational questions that are yet to be resolved in order to enable a mathematicallyrigorous and unambiguous interpretation of GW observations.

2. Perturbation MethodsContributor: B. Wardell

Exact models for GWs from BBHs can only be obtained by exactly solving the fullEinstein field equations. However, there is an important regime in which a perturbative


treatment yields a highly accurate approximation. For BBH systems in which one ofthe BHs is much less massive than the other, one may treat the mass ratio as a smallperturbation parameter. Then, the Einstein equations are amenable to a perturbativeexpansion in powers of this parameter. Such an expansion is particularly suitable forEMRIs, systems in which the mass ratio may be as small as 10−6, or even smaller[697–701]. In such cases, it has been established [702–704] that it will be necessaryto incorporate information at second-from-leading perturbative order to achieve theaccuracy that will be required for optimal parameter estimation by the planned LISAmission [672, 705–708]. Aside from EMRIs, a perturbative expansion is likely to alsobe useful as a model for Intermediate Mass Ratio Inspirals (IMRIs): systems where themass ratio may be as large as ∼ 10−2. Such systems, if they exist, are detectable inprinciple by Advanced LIGO and Virgo, and are indeed being looked for in the data ofthese experiments[709, 710].

The perturbative approach (often called the self-force∗ approach) yields a set ofequations for the motion of the smaller object about the larger one. For a detailedtechnical review of self-force physics, see [719], and for a most recent, pedagogicalreview, see [720]. At zeroth order in the expansion, one recovers the standard geodesicequations for a test particle in orbit around (i.e. moving in the background spacetime of)the larger BH. At first order, we obtain coupled equations for an accelerated worldlineforced off a geodesic by the GSF, which itself arises from the metric perturbation tothe background spacetime sourced by the stress-energy of the smaller BH. In the GSFapproach, it can be convenient to treat the smaller BH as a “point particle”, withthe GSF being computed from an effective regularized metric perturbation [721–725].Such an assumption is not strictly necessary, but has been validated by more carefultreatments whereby both BHs are allowed to be extended bodies, and the point-particle-plus-regularization prescription is recovered by appropriately allowing the smaller BHto shrink down to zero size [726–729]. These more careful treatments have also allowedthe GSF prescription to be extended to second perturbative order [729–735] and to thefully non-perturbative case [736–738].

The goal of the GSF approach is to develop efficient methods for computing themotion of the smaller object and the emitted GWs in astrophysically-relevant scenarios.These involve, most generally, a spinning (Kerr) large BH, and a small (possiblyspinning) compact object in a generic (possibly inclined and eccentric) inspiral orbitaround it. The data analysis goals of the LISA mission (which demand that the phaseof the extracted waveform be accurate to within a fraction of a radian over the entireinspiral) require all contributions to the metric perturbation at first order, along withthe dissipative contributions at second order [702].

∗ Strictly speaking, the term self-force refers to the case where local information about the perturbationin the vicinity of the smaller object is used; other calculations of, e.g., the flux of GW energy far fromthe binary also rely on a perturbative expansion [711–718], but are not referred to as GSF calculations.


2.1. Recent developments

Focused work on the GSF problem has been ongoing for at least the last twodecades, during which time there has been substantial progress. Early work developedmuch of the mathematical formalism, particularly in how one constructs a well-motivated and unambiguous regularized first-order metric perturbation [721–727, 739].More recent work has addressed the conceptual challenges around how these initialresults can be extended to second perturbative order [729–735], and on turning theformal mathematical prescriptions into practical numerical schemes [740–745]. As aresult, we are now at the point where first-order GSF calculations are possible foralmost any orbital configuration in a Kerr background spacetime [746]. Indeed, we nowhave not one, but three practical schemes for computing the regularized first-order GSF[747]. Some of the most recent highlights from the substantial body of work created bythe GSF community are discussed below.

2.1.1. First-order gravitational self-force for generic orbits in Kerr spacetime Progressin developing tools for GSF calculations has been incremental, starting out withthe simplest toy model of a particle with scalar charge in a circular orbit about aSchwarzschild BH, and then extending to eccentric, inclined, and generic orbits abouta spinning Kerr BH [748–775]. Progress has also been made towards developing toolsfor the conceptually similar, but computationally more challenging GSF problem; againstarting out with simple circular orbits in Schwarzschild spacetime before extending toeccentric equatorial orbits and, most recently, fully generic orbits in Kerr spacetime[743, 776–793]. Along the way, there have been many necessary detours in order toestablish the most appropriate choice of gauge [783, 789, 791, 794–798], reformulationsof the regularization procedure [769, 785, 789, 799–806], and various numerical methodsand computational optimisations [757, 758, 761, 781, 785, 788, 794, 795, 807–813].

2.1.2. Extraction of gauge-invariant information Both the regularized metricperturbation and the GSF associated with it are themselves gauge-dependent [796,814, 815], but their combination encapsulates gauge-invariant information. A seriesof works have derived a set of gauge-invariant quantities accessible from the regularizedmetric and GSF, which quantify conservative aspects of the dynamics in EMRI systemsbeyond the geodesic approximation. (Strictly speaking, they are only “gauge invariant”within a particular, physically motivated class of gauge transformations. Nevertheless,even with this restriction the gauge invariance is very useful for comparisons withother results.) Examples include Detweiler’s red-shift invariant [816], the frequencyof the innermost stable circular orbit (ISCO) in Schwarzschild [817] and Kerr [790], theperiastron advance of slightly eccentric orbits in Schwarzschild [818] and Kerr [819],spin (geodetic) precession [820–823], and, most recently, quadrupolar and octupolartidal invariants [824, 825]. The most important outcome from the development of thesegauge invariants is the synergy it has enabled, both within the GSF programme [826]


(e.g., by allowing for direct comparisons between results computed in different gauges)and, as described next, with other approaches to the two-body problem.

2.1.3. Synergy with PN approximations, EOB theory, and NR One of the most fruitfuloutcomes arising from the development of GSF gauge-invariants is the synergy it hasenabled between the GSF and PN and EOB theories. With a gauge-invariant descriptionof the physical problem available, it is possible to make direct connections between GSF,PN and EOB approximations. This synergy has worked in a bidirectional way: GSFcalculations have been used to determine previously-unknown coefficients in both PNand EOB expansions [816, 818, 827–841]; and EOB and PN calculations have been usedto validate GSF results [830], and even to assess the region of validity of the perturbativeapproximation [842–844]. More on this in Sec. 3.

Mirroring the synergy between GSF and PN/EOB, there have emerged methods formaking comparisons between GSF and NR. This started out with direct comparisonsof the periastron advance of slightly eccentric orbits [842, 844]. More recently, a similarcomparison was made possible for Detweiler’s redshift [845, 846], facilitated by anemerging understating of the relation between Detweiler’s red-shift and the horizonsurface gravity of the small BH.

2.1.4. New and efficient calculational approaches Despite the significant progressin developing numerical tools for computing the GSF, it is still a computationallychallenging problem, particularly in cases where high accuracy is required. Thischallenge has prompted the development of new and efficient calculational approachesto the problem.

Initial GSF results were obtained in the Lorenz gauge [777, 778, 781, 782, 784, 788,817, 847], where the regularization procedure is best understood. Unfortunately, thedetails of a Lorenz-gauge calculation—in which one must solve coupled equations forthe 10 components of the metric perturbation—are tedious and cumbersome, makingit difficult to implement and even more difficult to achieve good accuracy. In theSchwarzschild case, other calculations based on variations of Regge-Wheeler gauge[792, 794, 808–810, 812, 848, 849] were found to be much easier to implement and yieldedmuch more accurate results. However, with regularization in Regge-Wheeler gauge lesswell understood, those calculations were restricted to the computation of gauge-invariantquantities. Perhaps more importantly, they are restricted to Schwarzschild spacetime,meaning they can not be used in astrophysically realistic cases where the larger BH isspinning. The radiation gauge—in which one solves the Teukolsky equation [850, 851] fora single complex pseudo-scalar ψ4 (or, equivalently, ψ0)—retains much of the simplicityof the Regge-Wheeler gauge, but has the significant benefit of being capable of describingperturbations of a Kerr BH. Furthermore, recent progress has clarified subtle issuesrelated to regularization [789] (including metric completion [852, 853]) in radiationgauge, paving the way for high-accuracy calculations of both gauge-invariant quantitiesand the GSF [746, 819, 854, 855].


Within these last two approaches (using Regge-Wheeler and radiation gauges)functional methods [856–858] have emerged as a particularly efficient means of achievinghigh accuracy when computing the metric perturbation. Fundamentally, these methodsrely on the fact that solutions of the Teukolsky (or Regge-Wheeler) equation can bewritten as a convergent series of hypergeometric functions. This essentially reducesthe problem of computing the metric perturbation to the problem of evaluatinghypergeometric functions. The approach has proved very successful, enabling bothhighly accurate numerical calculations [746, 859–861] and even exact results in the low-frequency–large-radius (i.e. PN) regime [830, 832–834, 837–841, 862, 863]. A differenttype of analytic treatment is possible for modelling the radiation from the last stage ofinspiral into a nearly extremal BH, thanks to the enhanced conformal symmetry in thisscenario [864–867]. This method, based on a scheme of matched asymptotic expansions,has so far been applied to equatorial orbits [773, 868], with the GSF neglected.

2.1.5. Cosmic censorship Independently of the goal of producing accurate waveformsfor LISA data analysis, the GSF programme has also yielded several other importantresults. One particular area of interest has been in the relevance of the GSF to answeringquestions about cosmic censorship. Calculations based on test-particle motion madethe surprising discovery that a test particle falling into a Kerr BH had the potentialto increase the BH spin past the extremal limit, thus yielding a naked singularity[869]. (Analogous cases exist where an electric charge falling into a charged (Reissner-Nordström) BH may cause the charge on the BH to increase past the extremal limit[870–872].) The intuitive expectation is that this is an artifact of the test-particleapproximation, and that by including higher-order terms in this approximation theGSF may in effect act as a “cosmic censor” by preventing over-charging and restoringcosmic censorship [873]. Several works have explored this issue in detail, studying theself-force on electric charges falling into a Reissner-Nördstrom BH [874, 875] and ona massive particle falling into a Kerr BH [876, 877]. These works demonstrated withexplicit calculations how the overspinning or overcharging scenarios are averted oncethe full effect of the self-force is taken into account (a result later rigorously proven ina more general context [878]).

2.2. Remaining challenges and prospects

While the perturbative (self-force) approach has proven highly effective to date,there remain several important and challenging areas for further development. Here,we list a number of the most important future challenges and prospects for the GSFprogramme.

2.2.1. Efficient incorporation of self-force information into waveform models Thereare at least two key aspects to producing an EMRI model: (i) computing the GSF; and(ii) using the GSF to actually drive an inspiral. Unfortunately, despite the substantial


advances in calculational approaches, GSF calculations are still much too slow to beuseful on their own as a means for producing LISA gravitational waveforms. Existingwork has been able to produce first-order GSF driven inspirals for a small number ofcases [764, 784, 792, 879] but when one takes into account the large parameter spaceof EMRI systems it is clear that these existing methods are inadequate. It is thereforeimportant to develop efficient methods for incorporating GSF information into EMRImodels and waveforms. There has been some promising recent progress in this direction,with fast “kludge” codes producing approximate (but not sufficiently accurate) inspirals[711, 713, 716, 717, 880–882], and with the emergence of mathematical frameworksbased on near-identity transformations [883], renormalization group methods [884] andtwo-timescale expansions [885].

To complicate matters, inspirals generically go through a number of transientresonances, when the momentary radial and polar frequencies of the orbit occur ina small rational ratio. During such resonances, approximations based on adiabaticitybreak down [886–890]. Works so far have mapped the locations of resonances in theinspiral parameter space, studied how the orbital parameters (including energy andangular momentum) experience a “jump” upon resonant crossing, and illustrated howthe magnitude of the jump depends sensitively on the precise resonant phase (therelative phase between the radial and polar motions at resonance). The impact ofresonances on the detectability of EMRIs with LISA was studied in Refs. [891, 892]using an approximate model of the resonant crossing. But so far there has been noactual calculation of the orbital evolution through a resonance. Now that GSF codesfor generic orbits are finally at hand, such calculations become possible, in principle.There is a vital need to perform such calculations, in order to allow orbital evolutionmethods to safely pass through resonances without a significant loss in the accuracy ofthe inspiral model.

2.2.2. Producing accurate waveform models: self-consistent evolution and second-ordergravitational self-force Possibly the most challenging outstanding obstacle to reachingthe sub-radian phase accuracy required for LISA data analysis is the fact that thefirst-order GSF on its own is insufficient and one must also incorporate informationat second perturbative order [702]. There are ongoing efforts to develop tools forcomputing the second order GSF [727, 734, 735, 769, 806, 885, 893–896], but, despitesignificant progress, a full calculation of the second-order metric perturbation has yetto be completed.

One of the challenges of the GSF problem when considered through secondorder is that it is naturally formulated as a self-consistent problem, whereby thecoupled equations for the metric perturbation and for the particle worldline are evolvedsimultaneously. Indeed, this self-consistent evolution has yet to be completed even forthe first-order GSF (it has, however, been done for the toy-model scalar charge case[764]). Even when methods are developed for computing the second order GSF, it willremain a further challenge to incorporate this information into a self-consistent evolution


scheme.

2.2.3. Gravitational Green Function One of the first proposals for a practical methodfor computing the GSF was based on writing the regularized metric perturbation interms of a convolution, integrating the Green function for the wave equation along theworldline of the particle [740, 743]. It took several years for this idea to be turnedinto a complete calculation of the GSF, and even then the results were restricted tothe toy-model problem of a scalar charge moving in Schwarzschild spacetime [768, 771].Despite this deficiency, the Green function approach has produced a novel perspectiveon the GSF problem.

In principle, the methods used for a scalar field in Schwarzschild spacetime shouldbe applicable to the GSF problem in Kerr. The challenge is two-fold: (i) to actuallyadapt the methods to the Kerr problem and to the relevant wave equation, which is thelinearized Einstein equation in the Lorenz gauge; and (ii) to explore whether and howone can instead work with the much simpler Teukolsky wave equation.

2.2.4. Internal-structure effects The vast majority of GSF calculations to date havebeen based on the assumption that the smaller object is spherically symmetric andnon-spinning. This idealization ignores the possibility that the smaller BH may bespinning, or more generally that other internal-structure effects may be relevant.Unfortunately, this picture is inadequate; certain internal-structure effects can makeimportant contributions to the equations of motion. For example, the coupling of thesmall body’s spin to the larger BH’s spacetime curvature (commonly referred to asthe Mathisson-Papapetrou force) is expected to contribute to the phase evolution of atypical EMRI at the same order as the conservative piece of the first-order GSF [897].Furthermore, finite internal-structure is likely to be even more important in the casethat the smaller body is a NS.

While there has been some progress in assessing the contribution from the smallerbody’s spin to the motion [879, 897–903], existing work has focused on flux-basedcalculations or on the PN regime. It remains an outstanding challenge to determinethe influence of internal-structure effects on the GSF, especially at second order.

2.2.5. EMRIs in alternative theories of gravity In all of the discussion so far, we havemade one overarching assumption: that BHs behave as described by General Relativity(GR). However, with EMRIs we have the exciting prospect of not simply assuming thisfact, but of testing its validity with exquisite precision. There is initial work on theself-force in the context of scalar-tensor gravity [904], but much more remains to bedone to establish exactly what EMRIs can do to test the validity of GR (and how) whenpushed to its most extreme limits. Much more on this in Chapter III (see, in particular,Sec. 3.3 therein)


2.2.6. Open tools and datasets While there has been significant progress in developingtools for computing the GSF, much of it has been ad-hoc, with individual groupsdeveloping their own private tools and codes. Now that a clear picture has emergedof exactly which are the most useful methods and tools, the community has begun tocombine their efforts. This has lead to the development of a number of initiatives,including (i) tabulated results for Kerr quasinormal modes and their excitationfactors [905, 906]; (ii) open source “kludge” codes for generating an approximatewaveform for EMRIS [881, 907]; and (ii) online repositories of self-force results [908].It is important for such efforts to continue, so that the results of the many years ofdevelopment of GSF tools and methods are available to the widest possible user base.One promising initiative in this direction is the ongoing development of the Black HolePerturbation Toolkit [909], a free and open source set of codes and results produced bythe GSF community.

3. Post-Newtonian and Post-Minkowskian MethodsContributor: A. Le Tiec

3.1. Background

The PN formalism is an approximation method in GR that is well suited to describethe orbital motion and the GW emission from binary systems of compact objects, ina regime where the orbital velocity is small compared to the speed of light and thegravitational fields are weak. This approximation method has played a key role inthe recent detections, by the LIGO and Virgo observatories, of GWs generated byinspiralling and merging BH and NS binaries [1, 2, 11, 18, 21], by providing accuratetemplate waveforms to search for those signals and to interpret them. Here we give abrief overview of the application of the PN approximation to binary systems of compactobjects, focusing on recent developments and future prospects. See the review articles[910–917] and the textbooks [631, 918] for more information.

In PN theory, relativistic corrections to the Newtonian solution are incorporatedin a systematic manner into the equations of motion (EOM) and the radiation field,order by order in the small parameter v2/c2 ∼ Gm/(c2r), where v and r are the typicalrelative orbital velocity and binary separation, m is the sum of the component masses,and we used the fact that v2 ∼ Gm/r for bound motion. (The most promising sourcesfor current and future GW detectors are bound systems of compact objects.)

Another important approximation method is the post-Minkowskian (PM)approximation, or non-linearity expansion in Newton’s gravitational constant G, whichassumes weak fields (Gm/c2r 1) but unrestricted speeds (v2/c2 . 1), and perturbsabout the limit of special relativity. In fact, the construction of accurate gravitationalwaveforms for inspiralling compact binaries requires a combination of PN and PMtechniques in order to solve two coupled problems, namely the problem of motion and


that of wave generation.The two-body EOM have been derived in a PN framework using three well-

developed sets of techniques in classical GR: (i) the PN iteration of the Einsteinfield equations in harmonic coordinates [919–923], (ii) the Arnowitt-Deser-Misner(ADM) canonical Hamiltonian formalism [924–926], and (iii) a surface integral approachpioneered by Einstein, Infeld and Hoffmann [927–929]. By the early 2000s, each of theseapproaches has independently produced a computation of the EOM for binary systemsof non-spinning compact objects through the 3rd PN order (3PN).∗ More recently, theapplication of effective field theory (EFT) methods [930], inspired from quantum fieldtheory, has provided an additional independent derivation of the 3PN EOM [931]. Allof those results were shown to be in perfect agreement. Moreover, the 3.5PN terms—which constitute a 1PN relative correction to the leading radiation-reaction force—arealso known [932–937].

At the same time, the problem of radiation (i.e., computing the field in the far/wavezone) has been extensively investigated within the multipolar PM wave generationformalism of Blanchet and Damour [938–940], using the “direct integration of the relaxedEinstein equation” approach of Will, Wiseman and Pati [941, 942], and more recentlywith EFT techniques [943, 944]. The application of these formalisms to non-spinningcompact binaries has, so far, resulted in the computation of the GW phase up to therelative 3.5PN (resp. 3PN) order for quasi-circular (resp. quasi-eccentric) orbits [945–952], while amplitude corrections in the GW polarizations are known to 3PN order, andeven to 3.5PN order for the quadrupolar mode [953–958].

3.2. Recent developments

Over the last five years or so, significant progress on PN modelling of compactbinary systems has been achieved on multiple fronts, including (i) the extension ofthe EOM to 4PN order for non-spinning bodies (with partial results also obtained foraspects of the two-body dynamics at the 5PN order [818, 827, 959]), (ii) the inclusionof spin effects in the binary dynamics and waveform, (iii) the comparison of several PNpredictions to those from GSF theory, and (iv) the derivation of general laws controllingthe mechanics of compact binaries.

3.2.1. 4PN equations of motion for non-spinning compact-object binaries Recently,the computation of the two-body EOM has been extended to 4PN order, by using boththe canonical Hamiltonian framework in ADM-TT coordinates [960–965] and a FokkerLagrangian approach in harmonic coordinates [966–970]. Partial results at 4PN orderhave also been obtained using EFT techniques [971–973]. All of those high-order PNcalculations resort to a point-particle model for the (non-spinning) compact objects,and rely on dimensional regularization to treat the local ultraviolet (UV) divergences

∗ By convention, “nPN” refers to EOM terms that are O(1/c2n) smaller than the Newtonian acceleration,or, in the radiation field, smaller by that factor relative to the standard quadrupolar field.


that are associated with the use of point particles. The new 4PN results have been usedto inform the EOB framework [974], a semi-analytic model of the binary dynamics andwave emission (see Sec. 6 below).

The occurrence at the 4PN order of infrared (IR) divergences of spatial integralsled to the introduction of several ambiguity parameters; one in the ADM Hamiltonianapproach and two in the Fokker Lagrangian approach. One of those IR ambiguityparameters was initially fixed by requiring agreement with an analytical GSF calculation[975] of the so-called Detweiler redshift along circular orbits [816, 826]. Recently,however, Marchand et al. [969] gave the first complete (i.e., ambiguity-free) derivationof the 4PN EOM. The last remaining ambiguity parameter was determined from firstprinciples, by resorting to a matching between the near-zone and far-zone fields, togetherwith a computation of the conservative 4PN tail effect in d dimensions, allowing to treatboth UV and IR divergences using dimensional regularization.

Another interesting (and related) feature of the binary dynamics at the 4PN orderis that it becomes non-local in time [962, 966], because of the occurence of a GWtail effect at that order: gravitational radiation that gets scattered off the backgroundspacetime curvature backreacts on the orbital motion at later times, such that thebinary’s dynamics at a given moment in time depends on its entire past history [976–978].

3.2.2. Spin effects in the binary dynamics and gravitational waveform Since stellar-mass and/or supermassive BHs may carry significant spins [979, 980], much effort hasrecently been devoted to include spin effects in PN template waveforms. In particular,spin-orbit coupling terms linear in either of the two spins have been computed up tothe next-to-next-to leading order, corresponding to 3.5PN order in the EOM, usingthe ADM Hamiltonian framework [981, 982], the PN iteration of the Einstein fieldequations in harmonic coordinates [983, 984], and EFT techniques [985]. Spin-spincoupling terms proportional to the product of the two spins have also been computed tothe next-to-next-to leading order, corresponding to 4PN order in the EOM, using theADM Hamiltonian and EFT formalisms [986–990]. The leading order 3.5PN cubic-in-spin and 4PN quartic-in-spin contributions to the binary dynamics are also known forgeneric compact bodies [991–995], as well as all higher-order-in-spin contributions forBBHs (to leading PN order) [996]. All these results are summarized in Fig. 9. 2PN BHbinary spin precession was recently revisited using multi-timescale methods [997, 998],uncovering new phenomenology such as precessional instabilities [999] and nutationalresonances [1000].

Spin-related effects on the far-zone field have also been computed to high orders,for compact binaries on quasi-circular orbits. To linear order in the spins, those effectsare known up to the relative 4PN order in the GW energy flux and phasing [1001, 1002],and to 2PN in the wave polarizations [1003, 1004]. At quadratic order in the spins, thecontributions to the GW energy flux and phasing have been computed to 3PN order[1005, 1006], and partial results were derived for amplitude corrections to 2.5PN order


[1007]. The leading 3.5PN cubic-in-spin effects in the GW energy flux and phasing areknown as well [993].

Figure 9. Contributions to the two-body Hamiltonian in the PN spin expansion, forarbitrary-mass-ratio binaries with spin induced multipole moments. Contributions inred are yet to be calculated. LO stands for “leading order”, NLO for “next-to-leadingorder”, and so on. SO stands for “spin-orbit”. Figure from Ref. [996].

Finally, some recent works have uncovered remarkable relationships between the PN[996] and PM [1008] dynamics of a binary system of spinning BHs with an arbritrarymass ratio on the one hand, and that of a test BH in a Kerr background spacetime onthe other hand. Those results are especially relevant for the ongoing development ofEOB models for spinning BH binaries (see Sec. 6), and in fact give new insight into theenergy map at the core of such models.

3.2.3. Comparisons to perturbative gravitational self-force calculations The GWsgenerated by a coalescing compact binary system are not the only observable ofinterest. As we have described in Sec. 2, over recent years, several conservativeeffects on the orbital dynamics of compact-object binaries moving along quasi-circularorbits have been used to compare the predictions of the PN approximation to thoseof the GSF framework, by making use of gauge-invariant quantities such as (i) theDetweiler redshift [816, 827, 828, 831, 1009, 1010], (ii) the relativistic periastron advance[818, 819, 842, 844], (iii) the geodetic spin precession frequency [820], and (iv) varioustidal invariants [824, 825], all computed as functions of the circular-orbit frequency of


the binary. Some of these comparisons were extended to generic bound (eccentric)orbits [822, 836, 847]. All of those comparisons showed perfect agreement in thecommon domain of validity of the two approximation schemes, thus providing crucialtests for both methods. Building on recent progress on the second-order GSF problem[729, 734, 735, 796, 895, 896], we expect such comparisons to be extended to secondorder in the mass ratio, e.g. by using the redshift variable [893].

Independently, the BH perturbative techniques of Mano, Suzuki and Takasugi[857, 1011] have been applied to compute analytically, up to very high orders, the PNexpansions of the GSF contributions to the redshift for circular [838, 862, 1012, 1013]and eccentric [832, 833, 837, 839, 840] orbits, the geodetic spin precession frequency[841, 863, 1014], and various tidal invariants [821, 830, 834]. Additionally, using similartechniques, some of those quantities have been computed numerically, with very highaccuracy, allowing the extraction of the exact, analytical values of many PN coefficients[854, 859, 861].

3.2.4. First law of compact binary mechanics The conservative dynamics of a binarysystem of compact objects has a fundamental property now known as the first law ofbinary mechanics [1015]. Remarkably, this variational formula can be used to relate localphysical quantities that characterize each body (e.g. the redshift) to global quantitesthat characterize the binary system (e.g. the binding energy). For point-particle binariesmoving along circular orbits, this law is a particular case of a more general result, validfor systems of BHs and extended matter sources [1016].

Using the ADM Hamiltonian formalism, the first law of [1015] was generalizedto spinning point particles, for spins (anti-)aligned with the orbital angularmomentum [1017], and to non-spinning binaries moving along generic bound (eccentric)orbits [1018]. The derivation of the first law for eccentric motion was then extendedto account for the non-locality in time of the orbital dynamics due to the occurence atthe 4PN order of a GW tail effect [1019]. These various laws were derived on generalgrounds, assuming only that the conservative dynamics of the binary derives from anautonomous canonical Hamiltonian. (First-law-type relationships have also been derivedin the context of linear BH perturbation theory and the GSF framework [1020–1022].)Moreover, they have been checked to hold true up to 3PN order, and even up to 5PNorder for some logarithmic terms.

So far the first laws have been applied to (i) determine the numerical value ofthe aforementioned ambiguity parameter appearing in derivations of the 4PN two-bodyEOM [975], (ii) calculate the exact linear-in-the-mass-ratio contributions to the binary’sbinding energy and angular momentum for circular motion [843], (iii) compute theshift in the frequencies of the Schwarzschild and Kerr innermost stable circular orbitsinduced by the (conservative) GSF [782, 790, 817, 819, 829, 843, 1009], (iv) test theweak cosmic censorship conjecture in a scenario where a massive particle subject tothe GSF falls into a nonrotating BH along unbound orbits [876, 877], (v) calibratethe effective potentials that enter the EOB model for circular [829, 1023] and mildly


eccentric orbits [833, 835, 839], and spin-orbit couplings for spinning binaries [832], and(vi) define the analogue of the redshift of a particle for BHs in NR simulations, thusallowing further comparisons to PN and GSF calculations [845, 846].

3.3. Prospects

On the theoretical side, an important goal is to extend the knowledge of theGW phase to the relative 4.5PN accuracy, at least for non-spinning binaries on quasi-circular orbits. (Partial results for some specific tail-related effects were recently derived[1024].) This is essential in order to keep model systematics as a sub-dominant sourceof error when processing observed GW signals [1025]. Accomplishing that will require,in particular, the calculation of the mass-type quadrupole moment of the binary to 4PNorder and the current-type quadrupole and mass-type octupole moments to 3PN order.Moreover, some spin contributions to the waveform—both in the phasing and amplitudecorrections—still have to be computed to reach the 4PN level, especially at quadraticorder in the spins.

Most of the PN results reviewed here have been established for circularized binaries,and often for spins aligned or anti-aligned with the orbital angular momentum. It isimportant to extend this large body of work to generic, eccentric, precessing systems.Progress on the two-body scattering problem would also be desirable [1026–1028].Additionally, much of what has been achieved in the context of GR could be doneas well for well-motivated alternative theories of gravitation, such as for scalar-tensorgravity (e.g. [1029–1033]) or in quadratic gravity [1034, 1035].

The first law of binary mechanics reviewed above could be extended to generic,precessing spinning systems, and the effects of higher-order spin-induced multipolesshould be investigated. Recent work on the PM approximation applied to thegravitational dynamics of compact binaries has given new insight into the EOB model[1008, 1036–1038], and in particular into the energy map therein. These two lines ofresearch may improve our physical understanding of the general relativistic two-bodyproblem.

On the observational side, future GW detections from inspiralling compact-objectbinaries will allow testing GR in the strong-field/radiative regime, by constrainingpossible deviations from their GR values of the various PN coefficients that appearin the expression of the phase. This, in particular, can be used to test some importantnonlinear features of GR such as the GW tail, tail-of-tail and nonlinear memory effects.Indeed, the first detections of GWs from inspiralling BH binaries have already been usedto set bounds on these PN coefficients, including an O(10%) constraint on the leadingtail effect at the 1.5PN order [3]; see Fig. 10 (or Ref. [18] for a more up-to-date versionthereof). More detections with a wider network of increasingly senstive interferometricGW detectors will of course improve those bounds.

Finally, following the official selection of the LISA mission by the European SpaceAgency (ESA), with a launch planned for 2034, we foresee an increased level of activity


Figure 10. Two GW detections of inspiralling BH binaries were used to set boundson possible deviations from their GR values of various PN coefficients that appear inthe expression for the GW phase. Figure from Ref. [3].

in source modelling of binary systems of MBHs and EMRIs, two promising classes ofsources for a mHz GW antenna in space. This will motivate more work at the interfacebetween the PN approximation and GSF theory.

4. Numerical Relativity and the Astrophysics of Black Hole BinariesContributor: P. Schmidt

The year of 2005 marked a remarkable breakthrough: the first successful numericalsimulation of—and the extraction of the GWs from—an inspiraling pair of BHs throughtheir merger and final ringdown [1039–1041] (see e.g. [1042] for a review).

NR provides us with accurate gravitational waveforms as predicted by GR. BBHscover an eight-dimensional parameter space spanned by the mass ratio q = m1/m2,the spin angular momenta Si and the eccentricity e. Simulations are computationallyextremely expensive, thus the large BBH parameter space is still sparsely sampled.Nevertheless, NR waveforms already play a crucial part in the construction and


verification of semi-analytic waveform models used in GW searches, which facilitatedthe first observations of GWs from BBH mergers [1–3, 11, 18]. Furthermore, they playa key role in the estimation of source properties and in facilitating important tests ofGR in its most extreme dynamical regime.

Since the initial breakthrough, NR has made significant progress: from the firstsimulations of equal-mass non-spinning BBHs spanning only the last few orbits [1039–1041], to a realm of simulations exploring aligned-spin [1043–1045] as well as precessingquasi-circular binaries [1046–1048], eccentric-orbit binaries [1049, 1050], and evolutionslong enough to reach into the early-inspiral regime where they can be matched onto PNmodels [1051].

Today, several codes are capable of stably evolving BBHs and extracting theirGW signal. They can roughly be divided into two categories: finite-differencing codesincluding BAM [1052], the Einstein Toolkit [1053], LazEv [623, 1040, 1043, 1054],MAYA [1055], LEAN [1056] as well as the codes described in Refs. [1039, 1057, 1058];and (pseudo-)spectral codes such as the Spectral Einstein Code (SpEC) [1059]. Otherevolution codes currently under development include [1060] and [1061]. To date, thesecodes have together produced several thousands of BBH simulations [1062–1065].

4.1. Current status

Excision and puncture. In BBH simulations one numerically solves the vacuumEinstein equations with initial conditions that approximate a pair of separate BHs atsome initial moment. An obvious complication is the presence of spacetime singularitiesinside the BHs, where the solution diverges. There are two approaches to this problem.The first is excision [1066], whereby a region around the singularity is excised (removed)from the numerical domain. As no information can propagate outwards from the interiorof the BH, the physical content of the numerical solution outside the BHs is unaffected.Since the excision boundary is spacelike (not timelike), one cannot and does not specifyboundary conditions on it. Instead, the main technical challenge lies in ensuring that theexcision boundary remains spacelike. The risk, for example, is that part of the boundarymay become timelike if numerical noise isn’t properly controlled [1067–1069]. It mustalso be ensured that non-physical gauge modes do not lead to numerical instabilities.Excision is used in SpEC, for example.

The second common way to deal with the BH singularities is to choose singularity-avoiding coordinates. This is achieved by representing BHs as compactified topologicalwormholes [1070] or infinitely long cylinders (“trumpets”) [1071], known as punctureinitial data. Specific gauge conditions allow the punctures to move across the numericalgrid, giving this approach the name moving punctures [1040, 1041, 1071–1073].

Initial Data. No exact solutions are known, in general, for the BBH metric on theinitial spatial surface, so one resorts to approximate initial conditions. Two types ofinitial data are commonly used: conformally flat and conformally curved. Most simu-lations that incorporate moving punctures use conformally flat initial data. Under this


assumption, three of the four constraint equations (themselves a subset of the full Ein-stein field equations) are given analytically in terms of the Bowen-York solutions [1074].The maximal possible angular momentum in this approach is a/m = 0.93, known asthe Bowen-York limit [1075]. In order to go beyond this limit, conformally curved ini-tial data have to be constructed. For codes that use excision, these can be obtainedby solving the extended conformal thin sandwich (CTS) equations [1076] with quasi-elliptical boundary conditions [1077–1080]. The initial spatial metric is proportional toa superposition of the metrics of two boosted Kerr-Schild BHs [1081]. More recently,the first non-conformally flat initial data within the moving punctures framework havebeen constructed [1082, 1083] by superposing the metrics and extrinsic curvatures oftwo Lorentz-boosted, conformally Kerr BHs.

Evolution Systems. The successful evolution of a BBH spacetime further requiresa numerically stable formulation of the Einstein field equations and appropriate gaugechoices. Long-term stable evolutions today are most commonly performed with eithera variant of the generalised harmonic [1039, 1084, 1085] or the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation [1086, 1087]. Another formulation, Z4, com-bines constraint preserving boundary conditions with an evolution system very closeto BSSN [1088–1091]. More advanced Z4-type formulations, which are conformal andtraceless, were developed in Refs. [1092–1094]

Gravitational Wave Extraction. The GW signal emitted through the inspiral,merger and ringdown is usually extracted from the Newman-Penrose curvature scalarΨ4 [1095, 1096] associated with the computed metric. The extraction of the signal istypically performed on spheres of constant coordinate radius some distance from thebinary, followed by extrapolation to infinity. The method of Cauchy-CharacteristicExtraction (CCE) [1097, 1098] allows us to extract the observable GW signal directlyat future null infinity by matching the Cauchy evolution onto a characteristic evolutionthat extends the simulation to null infinity. In the Cauchy-Characteristic Extraction, thegravitational waveform is most naturally extracted from the Bondi news function [1099,1100].

4.2. Challenges

High spins. Numerical simulations of close to maximally spinning BHs are stillchallenging to carry out due to difficulties in the construction of initial data as well asincreasingly demanding accuracy requirements during the evolutions. This particularlyaffects binary configurations of unequal masses and arbitrary spin orientation, despitesignificant developments made in the past five years [1062–1065, 1101]. While spins upto 0.994 have been evolved stably [1102], extensive work is still underway to reach theproposed Novikov-Thorne limit of 0.998 [1103] and beyond.


High mass ratios. The highest mass ratio BBH fully general-relativistic numericalsimulation available to date is of q = 100 [1104]. (However, this simulation follows arelatively small number of orbital cycles.) For spinning BHs, simulations of mass ratioq = 18 have been performed [1101]. Both of these have been obtained in the mov-ing punctures approach. For spectral methods, mass ratios higher than q ∼ 10 arenumerically very challenging. Generally, higher mass ratios are more computationallyexpensive as the disparate lengthscales demand higher spatial resolution, and becausethe number of orbital cycles increases in proportion to q. The choice of a Courant factorposes another problem in explicit evolution schemes, as it forces a small time step, whichbecomes increasingly challenging for high mass ratios. Implicit schemes could provide apotential solution to this [1105], but have not yet been applied to the fully relativisticbinary problem. Improvements in numerical technique, and perhaps a synergy with theperturbative methods reviewed in Sec. 2, will be crucial for overcoming this problem inthe hope of extending the reach of NR towards the extreme-mass-ratio regime.

Long simulations. It is crucial for simulations to track the binary evolution fromthe early inspiral, where it can be matched to a PN model, all the way down to thefinal merger and ringdown. However, such long evolutions are computationally veryexpensive, requiring many months of CPU time with existing codes. Note that the runtime is not simply linear in the evolution time: a longer run would usually also requirea larger spatial domain in order to keep spatial boundaries out of causal contact. Todate, only one complete numerical simulation reaching well into the PN regime has beenproduced, using a modified version of SpEC [1051]. Significant modifications to existingcodes would need to be made in order to be able to generate long simulations on aproduction scale.

Waveform accuracy. As the sensitivity of GW detectors increases, further workwill be required to further improve models and assess their systematic errors [1025].Already, for large precessing spins, or large mass ratios, or non-negligible eccentricities,systemic modelling errors limit the accuracy with which LIGO and Virgo can extractsource parameters (see [1106] for a more detailed discussion). This is due, in part, torelatively large errors in current models of high multipole-mode contributions, which weexpect in the coming years to become resolvable in detected signals.

Beyond General Relativity. GW observations from merging compact binaries allowus to probe the strong-field regime and test GR (cf. Chapter III). While theory-agnostic ways to test deviations from GR are commonly used, testing a selection ofwell-motivated alternative theories through direct waveform comparison is desirable.First steps have been taken to simulate BBHs in alternative theories of gravity thatadmit BH solutions [1107, 1108]. However, many issues remain to be addressed, notleast of which the fact that some of these theories may not even possess a well-posedinitial value formulation (see [1109, 1110] for examples).


4.3. Numerical Relativity and GW observations

NR plays an important role in GW astrophysics and data analysis. The semi-analytical waveform models employed to search for BBHs (see Sec. 6 below) model thecomplete radiative evolution from the early inspiral, through the merger and to the finalringdown stage, with the later stages calibrated to BBH simulations [1111, 1112]. Thesemodels and more sophisticated ones incorporating more physical effects, for exampleprecession [1113, 1114], are used to determine the fundamental properties of the GWsource, i.e. its masses and spin angular momenta as well as extrinsic quantities such asthe orbital orientation relative to the observatories [1115, 1116].

Alternatively, pure NR waveforms may be used directly [1117, 1118] or by the meansof surrogate models [1119–1121]. But due to the computational cost of simulations, thesehave only been attempted so far on a very restricted portion of the parameter space.Despite this restriction, pure NR surrogate models have the advantage of incorporatingmore physical effects that may be limited or entirely neglected in currently availablesemi-analytic models.

The ringdown phase after the merger may be described by perturbation theory (seee.g. [1122] for a review), but the amplitudes of the excited quasi-normal modes can onlybe obtained from numerical simulations. These are of particular interest as measuringthe amplitudes of individual quasinormal modes would allow to map the final state ofthe merger to the properties of the progenitor BHs [1123, 1124] and to test the BHnature of the source [1125] (see also Sections 4.1 and 4.2).

In order to estimate the mass and spin angular momentum of the remnant BH, fitsto NR simulations are essential [1092–1094, 1126–1133]. Independent measurements ofthe binary properties from the inspiral portion of the GW signal and from the laterstages of the binary evolution using fits from NR allow, when combined, to test thepredictions from GR [17, 1134]. While the phenomenological fit formulae have seenmuch improvement in recent years, modelling the final state of precessing BH binariesfrom numerical simulations still remains an open challenge.

Generically, when BHs merge, anisotropic emission of GWs leads to the build upof a net linear momentum. Due to the conservation of momentum, once the GWemission subsides, the remnant BH recoils (“kick”). While the recoil builds up duringthe entire binary evolution, it is largest during the non-linear merger phase. Kickvelocities can be as high as several thousands km/s, with astrophysical consequences: aBH whose recoil velocity is larger than the escape velocity of its galaxy may leave itshost. Numerical simulations are necessary to predict the recoil velocities [1135–1138] (aconvenient surrogate model for these velocities was constructed recently in [1139]). Ithas been suggested that it may be possible to directly measure the recoil speed from theGW signal by observing the induced differential Doppler shift throughout the inspiral,merger and ringdown [1140].

Numerical simulations of the strong field regime are also crucial for exploring otherspin-related phenomena such as “spin-flips” [1141] caused by spin-spin coupling, whose


signatures may be observable. Understanding the spin evolution and correctly modellingspin effects is crucial for mapping out the spin distribution of astrophysical BHs fromGW observations.

Waveform models as well as fitting formulae for remnant properties areprone to systematic modelling errors. For GW observations with low SNR, thestatistical uncertainty dominates over the systematic modelling error in the parametermeasurement accuracy. Improved sensitivities for current and future GW observatories(including, especially, LISA [1142, 1143]) will allow for high SNR observations reachinginto the regime where systematic errors are accuracy limiting. This includes stronglyinclined systems, higher-order modes, eccentricity, precession and kicks, all of which canbe modelled more accurately through the inclusion of results from NR (see [1025] for adetailed discussion in the context of the first BBH observation GW150914).

5. Numerical relativity in fundamental physicsContributor: U. Sperhake

The standard model of particle physics and Einstein’s theory of GR provide uswith an exquisite theoretical framework to understand much of what we observe in theUniverse. From high-energy collisions at particle colliders to planetary motion, the GWsymphony of NS and BH binaries and the cosmological evolution of the Universe at large,the theoretical models give us remarkably accurate descriptions. And yet, there are gapsin this picture that prompt us to believe that something is wrong or incomplete. Galacticrotation curves, strong gravitational lensing effects, X-ray observations of galactic halosand the cosmic microwave background cannot be explained in terms of the expectedgravitational effects of the visible matter [283]. Either we are prepared to acceptthe need to modify the laws of gravity, or there exists a form of dark matter (DM)that at present we can not explain satisfactorily with the SM (or both). DifferentDM candidates and their status are reviewed in Section 5.1. Likewise, the acceleratedexpansion of the Universe [1144] calls for an exotic form of matter dubbed dark energyor (mathematically equivalently) the introduction of a cosmological constant with avalue many orders of magnitude below the zero-point energy estimated by quantumfield theory—the cosmological constant problem. Further chinks in the armor of theSM+GR model of the universe include the hierarchy problem, i.e. the extreme weaknessof gravity relative to the other forces, and the seeming irreconcilability of GR withquantum theory.

Clearly, gravity is at the center of some of the most profound contemporary puzzlesin physics. But it has now also given us a new observational handle on these puzzles, inthe form of GWs [1]. Furthermore, the aforementioned 2005 breakthrough in NR [1039–1041] has given us the tools needed to systematically explore the non-linear strong-field regime of gravity. For much of its history, NR was motivated by the modeling ofastrophysical sources of GWs, as we have described in Sec. 4. As early as 1992, however,Choptuik’s milestone discovery of critical behaviour in gravitational collapse [1145] has


demonstrated the enormous potential of NR as a tool for exploring a much wider rangeof gravitational phenomena. In this section we review the discoveries made in this fieldand highlight the key challenges and goals for future work.

5.1. Particle laboratories in outer space

DM, by its very definition, generates little if any electromagnetic radiation; rather,it interacts with its environment through gravity. Many DM candidates have beensuggested (see also Section 5), ranging from primordial BH clouds to weakly interactingmassive particles (WIMPs) and ultralight bosonic fields [283, 284, 1146]. The latter area particularly attractive candidate in the context of BH and GW physics due to theirspecific interaction with BHs. Ultralight fields, also referred to as weakly interactingslim particles (WISPs), typically have mass parameters orders of magnitude below anelectron volt and arise in extensions of the SM of particle physics or extra dimensions instring theory. These include axions or axion-like particles, dark photons, non-topologicalsolitons (so-called Q-balls) and condensations of bosonic states [283, 1147–1149]. Forreference, we note that a mass 10−10 eV (10−20 eV) corresponds to a Compton wavelengthof O(1) km (O(102) AU) which covers the range of Schwarzschild radii of astrophysicalBHs.

At first glance, one might think the interaction between such fundamental fields andBHs is simple; stationary states are constrained by the no-hair theorems and the fieldeither falls into the BH or disperses away. In practice, however, the situation is morecomplex—and more exciting. NR simulations have illustrated the existence of long-lived,nearly periodic states of single, massive scalar or Proca (i.e. vector) fields around BHs[1150–1152]. Intriguingly, these configurations are able to extract rotational energyfrom spinning BHs through superradiance [1153–1156], an effect akin to the Penroseprocess [1156, 1157]. Further details on this process are provided in Section 5.7. Inthe presence of a confining mechanism that prevents the field from escaping to infinity,this may even lead to a runaway instability dubbed the BH bomb [1158, 1159]. Anotherpeculiar consequence arising in the same context is the possibility of floating orbits wheredissipation of energy through GW emission is compensated by energy gain throughsuperradiance [1160, 1161].

Naturally, non-linear effects will limit the growth of the field amplitude or thelifetime of floating orbits, and recent years have seen the first numerical studies toexplore the role of non-linearities in superradiance. These simulations have shown thatmassive, real scalar fields around BHs can become trapped inside a potential barrieroutside the horizon, form a bound state and may grow due to superradiance [1150].Furthermore, beating phenomena result in a more complex structure in the evolutionof the scalar field. These findings were confirmed in [1151], which also demonstratedthat the scalar clouds can source GW emission over long timescales. The amplificationof GWs through superradiance around a BH spinning close to extremality has beenmodeled in Ref. [1162] and found to be maximal if the peak frequency of the wave


is above the superradiant threshold but close to the dominant quasi-normal modefrequency of the BH. The generation of GW templates for the inspiral and mergerof hairy BHs and the identification of possible smoking gun effects distinguishing themfrom their vacuum counterparts remains a key challenge for future NR simulations.

Numerical studies of the non-linear saturation of superradiance are very challengingdue to the long time scales involved. A particularly convenient example of superradiancein spherical symmetry arises through the interaction of charged, massless scalar fieldswith Reissner-Nordström BHs in asymptotically anti-de Sitter spacetime or in a cavity;here energy is extracted from the BH charge rather than its rotation. The scalarfield initially grows in accordance with superradiance, but eventually saturates, leavingbehind a stable hairy BH [1163, 1164]. Recently the instability of spinning BHs in AdSbackgrounds was studied in full generality [1165]. The system displays extremely richdynamics and it is unclear if there is a stationary final state.

The superradiant growth of a complex, massive vector field around a near-extremalKerr BH has recently been modeled in axisymmetry [1166]. Over 9 % of the BH masscan be extracted until the process gradually saturates. Massive scalar fields can also pileup at the center of “normal” stars. Due to gravitational cooling, this pile-up does notlead to BH formation but stable configurations composed of the star with a “breathing”scalar field [1167].

The hairy BH configurations considered so far are long-lived but not strictlystationary. A class of genuinely stationary hairy BH spacetimes with scalar or vectorfields has been identified in a series of papers [1168–1172]. The main characteristicof these systems is that the scalar field is not stationary—thus bypassing the no-hairtheorems—but the spacetime metric and energy-momentum are. A subclass of thesesolutions smoothly interpolates between the Kerr metric and boson stars. A majorchallenge for numerical explorations is to evaluate whether these solutions are non-linearly stable and might thus be astrophysically viable. A recent study of linearizedperturbations around the hairy BH solution of [1168] has found unstable modes withcharacteristic growth rates similar to or larger than those of a massive scalar field ona fixed Kerr background. However, such solution may still be of some astrophysicalrelevance [1173]. See also Sec. 4.2.1 of Chapter III for a related discussion.

5.2. Boson stars

The idea of stationary localized, soliton-like configurations made up of thetype of fundamental fields discussed in the previous section goes back to Wheeler’s“gravitational-electromagnetic entities” or geons of the 1950s [1174]. While Wheeler’ssolutions turn out to be unstable, replacing the electromagnetic field with a complexscalar field leads to “Klein-Gordon geons” first discovered by Kaup [1175] and now morecommonly referred to as boson stars. The simplest type of boson stars, i.e. stationarysolutions to the Einstein-complex-Klein-Gordon system, is obtained for a non-self-interacting field with harmonic time dependence φ ∝ eiωt where the potential contains


only a mass term V (φ) = m2|φ|2. The resulting one-parameter family of these so-calledmini boson star solutions is characterized by the central amplitude of the scalar field andleads to a mass-radius diagram qualitatively similar to that of static NSs; a maximummass value ofMmax = 0.633M2

Planck/m separates the stable und unstable branches [1176–1178]. For a particle mass m = 30 GeV, for instance, one obtains Mmax ∼ 1010 kg withradius R ∼ 6× 10−18m and a density 1048 times that of a NS [1179].

A wider range of boson star models is obtained by adding self-interaction termsto the potential V (φ) which result in more astrophysically relevant bulk properties ofthe stars. For a quartic term λ|φ|4/4, for example, the maximal mass is given byMPlanck(0.1 GeV2)M λ1/2/m2 [1180]; for further types of boson stars with differentpotential terms see the reviews [1178, 1179, 1181] and references therein. A particularlyintriguing feature of rotating boson stars exemplifies their macroscopic quantum-likenature: the ratio of angular momentum to the conserved particle number must beof integer value which prevents a continuous transition from rotating to non-rotatingconfigurations [1179, 1182]. More recently, stationary, soliton-like configurations havealso been found for complex, massive Proca fields [1183]. For real scalar fields,in contrast, stationary solutions do not exist, but localized, periodically oscillatingsolutions dubbed oscillatons have been identified in Ref. [1184].

Boson stars are natural candidates for DM [1178], but may also act as BH mimickers[1185, 1186]. In the new era of GW observations, it is vital to understand the GWgeneration in boson-star binaries and search for specific signatures that may enableus to distinguish them from BH or NS systems. Recent perturbative calculations ofthe tidal deformation of boson stars demonstrate that the inspiral part of the waveformmay allow us to discriminate boson stars, at least with third-generation detectors [1187].Numerical studies of dynamic boson stars have so far mostly focused on the stabilityproperties of single stars and confirmed the stable and unstable nature of the brancheseither side of the maximum mass configuration; see e.g. [1188–1191]. The modeling ofhead-on collisions of boson stars [1192, 1193] reveals rich structure in the scalar radiationand that the merger leads to the formation of another boson star. Head-on collisionshave also served as a testbed for confirming the validity of the hoop conjecture inhigh-energy collisions [1194]. Inspiralling configurations result either in BH formation,dispersion of the scalar field to infinity or non-rotating stars [1195, 1196], possibly aconsequence of the quantized nature of the angular momentum that makes it difficultto form spinning boson stars instead.

Binary boson star systems thus remain largely uncharted territory, especiallyregarding the calculation of waveform templates for use in GW data analysis and thequantized nature of spinning boson stars and their potential constraints on formingrotating stars through mergers.


5.3. Compact objects in modified theories of gravity

New physical phenomena and the signature of new “modified” theories are typicallyencountered when probing extreme regimes not accessible to previous experimentsand observations. Quantum effects, for instance, become prominent on microscopicscales and their observation led to the formulation of quantum theory while classicalphysics still provides an accurate description of macroscopic systems. Likewise,Galilean invariance and Newtonian theory accurately describe slow motion and weaklygravitating systems but break down at velocities comparable to the speed of light or inthe regime of strong gravity. We therefore expect modifications of GR, if present, toreveal themselves in the study of extreme scales such as the large-scale dynamics of theuniverse or the strong curvature regime near compact objects.

The dawn of GW observations provides us with unprecedented opportunities toprobe such effects. The modeling of compact objects in alternative theories of gravityrepresents one of the most important challenges for present and future NR studies, sothat theoretical predictions can be confronted with observations. This challenge facesadditional mathematical, numerical and conceptual challenges as compared to the GRcase; a more detailed discussion of these is given in Sec. 3.4 below. A convenient wayto classify the numerous modified theories of gravity is provided by the assumptionsunderlying Lovelock’s theorem and, more specifically, which of these assumptionsare dropped [1197]. Unfortunately, for most of these candidate theories, well-posedformulations are not known (cf. Table 1 in [1197]) or presently available only in theform of a continuous limit to GR or linearization around some background [1108, 1110].Prominent exceptions are (single- or multi-) scalar-tensor (ST) theories of gravity [1198]which includes Brans-Dicke theory [1199] and, through mathematical equivalence, asubset of f(R) theories [1200]. ST theories inherit the well-posedness of GR throughthe Einstein frame formulation [1201]; see also [1202–1204]. Furthermore, ST physicswould present the most conspicuous strong-field deviation from GR discovered so far, thespontaneous scalarization of NSs [1205] (for a similar effect in vector-tensor theory see[1206]). For these reasons, almost all NR studies have focused on this class of theories,even though its parameter space is significantly constrained by solar system tests andbinary pulsar observations [1207].

The structure of equilibrium models of NSs in ST theories has been studiedextensively in the literature (e.g. [1205, 1208–1214]) and leads to a mass-radius diagramthat contains one branch of GR or GR-like models plus possible additional branches ofstrongly scalarized stars. The presence or absence of these additional branches dependson the coupling between the scalar and tensor sector of the theory.

Early numerical studies considered the collapse of dust in spherical symmetry [1215–1218] which leads to a hairless BH in agreement with the no-hair theorems, even thoughdepartures from GR are possible during the dynamic stages of the collapse. In a sequenceof papers, Novak et al studied the collapse of NSs into a BH [1219], the transition ofNSs between stable branches [1220] and the formation of NSs through gravitational


collapse [1221]. In all cases, strong scalar radiation is generated for that part of theST theory’s parameter space that admits spontaneously scalarized equilibrium models.In [1222], the collapse of stellar cores to BHs was found to be the most promisingscenario to generate detectable scalar radiation for parameters allowed by the Cassiniand pulsar observations; galactic sources at a distance D = 10 kpc may be detectedwith present and future GW observatories or used to further constrain the theory’sparameters. All these simulations, however, consider massless ST theory. For massivefields, low frequency interactions decay exponentially with distance, so that the pulsarand Cassini constraints may no longer apply [1223]. In consequence, massive ST theorystill allows for very strongly scalarized equilibrium stars if m & 10−15 eV [1213, 1214].This has dramatic consequences for the GW signals that can be generated in massiveST theory as compared with its massless counterpart: GW amplitudes can be ordersof magnitude larger and the waves are spread out into a nearly monochromatic signalover years or even centuries due to the dispersion of the massive field. GW searchesmay therefore be directed at historic supernovae such as SN1987A and either observe asignal or constrain the theory’s parameter space [1224].

Numerical studies of binary systems in ST theory are rather scarce. The no-hair theorems strongly constrain possible deviations of pure BH spacetimes from GR.They can be bypassed, however, through non-trivial potentials [1225] or boundaryconditions [1107] which leads to scalar wave emission. Nevertheless, NS systems appearto be the more natural candidate to search for imprints of ST theory. Dynamicalscalarization has indeed been observed in simulations of the merger of two NSs withinitially vanishing scalar charge [1226, 1227]. Beyond GR and ST theory, we are onlyaware of one numerical study [1108], which simulated the evolution of BH binaries inthe dynamical Chern-Simons (dCS) theory linearized around GR. LIGO observationsmay then measure or constrain the dCS length scale to . O(10) km.

With the dawn of GW astronomy [1], the topics discussed so far in this sectionare becoming important subjects of observational studies with LIGO, Virgo and futureGW detectors. These studies are still in an early stage and the generation of precisionwaveforms for scenarios involving modifications of gravity, fundamental fields or moreexotic compact objects will be a key requirement for fully exploiting the scientificpotential of this new channel to observing the universe.

The analysis of GW events has so far concentrated on testing the consistency ofthe observed signals with GR predictions, establishing bounds on phenomenologicalparametrizations of deviations from GR and obtaining constraints from the propagationof the GW signal. An extended study of GW150914 demonstrated consistency betweenthe merger remnant’s mass and spins obtained separately from the low-frequency inspiraland the high-frequency postinspiral signal [17]. The Compton wavelength of the gravitonwas constrained with a 90% lower bound of 1013 km (corresponding to an upper boundfor the mass of ∼ 10−22 eV) and parametrizations of violations of GR using high-orderPN terms have been constrained; see also [11, 20]. The first NSB observation [21],combined with electromagnetic observations, limited the difference between the speed


of propagation of GWs and that of light to within −3× 10−15 and 7× 10−16 times thespeed of light [23]. Analysis of the polarization of the first triple coincidence detectionGW170814 found Bayes’ factors of 200 (1000) favoring a purely tensor polarizationagainst purely vector (purely scalar) radiation. In summary, the GW observations haveas yet not identified an inconsistency with the predictions of vacuum GR for BBH signalsor GR predictions for NS systems.

Chapter III contains detailed discussions on many of the topics raised here,including motivation and brief description of certain broad classes of alternative to GR(Sec. 2.1), numerics beyond GR (Sec. 3.4), and the nature of compact objects beyondGR (Sec. 4).

5.4. High-energy collisions of black holes

The hierarchy problem of physics consists in the vast discrepancy between theweak coupling scale (≈ 246 GeV) and the Planck scale 1.31× 1019 GeV or, equivalently,the relative weakness of gravity compared with the other interactions. A possibleexplanation has been suggested in the form of “large” extra spatial dimensions[1228, 1229] or extra dimensions with a warp factor [1230, 1231]. On short lengthscales. 10−4 m, gravity is at present poorly constrained by experiment and would, accordingto these models, be diluted due to the steeper fall off in higher dimensions. All otherinteractions, on the other hand, would be constrained to a 3+1 dimensional braneand, hence, be unaffected. In these braneworld scenarios, the fundamental Planckscale would be much smaller than the four-dimensional value quoted above, possiblyas low as O(TeV) which inspired the name TeV gravity. This fundamental Planck scaledetermines the energy regime where gravity starts dominating the interaction, leadingto the exciting possibility that BHs may be formed in particle collisions at the LHC orin high-energy cosmic rays hitting the atmosphere [1232–1234].

The analysis of experimental data employs so-called Monte-Carlo event generators[1235] which require as input the cross section for BH formation and the loss of energyand momentum through GW emission. In the ultrarelativistic limit, the particles maybe modeled as pointlike or, in GR, as BHs.

In D = 4 spacetime dimensions, high-energy collisions of BHs are by now wellunderstood. Head-on collisions near the speed of light radiate about 14 % of the center-of-mass energyM in GWs [1236, 1237]. The impact parameter separating merging fromscattering collisions with boost velocity v is b/M = (2.50 ± 0.05)/v [1238]. Grazingcollisions exhibit zoom-whirl behaviour [1239, 1240] and can radiate up to ∼ 50 % ofthe total energy in GWs [1241]. The collision dynamics furthermore become insensitiveto the structure of the colliding objects – BHs or matter balls – at high velocities[1194, 1241–1244] as expected when kinetic energy dominates the budget. Finally, NRsimulations of hyperbolic BH encounters are in good agreement with predictions by theEOB method [1026].

The key challenges are to generalize these results to the higher D scenarios relevant


for TeV gravity (there are partial, perturbative results [1245–1247]). Considering thesymmetries of the collision experiments, it appears plausible to employ rotationalsymmetry in higher D numerics which is vital to keep the computational costs undercontrol and underlies all NR studies performed so far [1058, 1248–1251]. Nonetheless,NR in higher D faces new challenges. The extraction of GWs is more complex, butnow tractable either with perturbative methods [1252–1254], using the Landau-Lifshitzpseudotensor [1255] or projections of the Weyl tensor [1256, 1257]. Likewise, initial datacan be obtained by generalizing four-dimensional techniques [1258]. Studies performedso far, however, indicate that, for reasons not yet fully understood, obtaining numericallystable evolutions is harder than in D = 4 [1259, 1260]. Results for the scatteringthreshold are limited to v . 0.5 c [1259] and the emission of GWs has only been computedfor non-boosted collisions of equal and unequal mass BH binaries [1260–1262]. Thesesimulations show a strong suppression of the radiated energy with D beyond its peakvalue Erad/M ≈ 9 × 10−4 at D = 5 for equal-mass systems, but reveal more complexbehaviour for low mass ratios. A remarkable outcome of BH grazing collisions in D = 5is the possibility of super-Planckian curvature in a region outside the BH horizons [1259].

5.5. Fundamental properties of black holes and non-asymptotically flat spacetimes

Recent years have seen a surge of NR applications to non-asymptotically flatspacetimes in the context of the gauge-gravity duality, cosmological settings and for theexploration of fundamental properties of BH spacetimes. We list here a brief selection ofsome results and open questions; more details can be found in the reviews [1263, 1264].

Cosmic censorship has for a long time been a topic of interest in NR, but to dateno generic, regular and asymptotically flat class of initial data are known to result inthe formation of naked singularities in four spacetime dimensions. Higher-dimensionalBHs, however, have a much richer phenomenology [1265], including in particular blackrings which may be subject to the Gregory-Laflamme instability [1266]. Thin black ringshave indeed been found to cascade to a chain of nearly circular BHs connected by everthinner segments in finite time [1267] in the same way as infinite black strings [1268].Similarly, ultraspinning topologically spherical BHs in D ≥ 6 dimensions are unstable[1269] and ultimately form ever thinner rings in violation of the weak cosmic censorshipconjecture [1270].

We have already mentioned Choptuik’s discovery of critical phenomena in thecollapse of spherical scalar fields [1145]. In asymptotically Anti-de Sitter (AdS)spacetimes, the dynamics change through the confining mechanism of the AdS boundary,allowing the scalar field to recollapse again and again until a BH forms [1271–1275]; see also [1276] for non-spherically symmetric configurations. NR simulations ofasymptotically AdS spacetimes are very challenging due to the complex outer boundaryconditions, in particular away from spherical symmetry, but recent years have seen thefirst simulations of BH collisions in AdS [1277, 1278] which, assuming gauge-gravityduality, would imply a far-from hydrodynamic behaviour in heavy-ion collisions during


the early time after merger.Using a cosmological constant with an opposite sign leads to asymptotically de

Sitter (dS) spacetimes widely believed to describe the Universe we live in. NR studiesin dS have explored the possible impact of local structures on the cosmological expansion[1279–1283] and found such inhomogeneities to not significantly affect the globalexpansion. Further work has explored the robustness of inflation under inhomogeneities[1284], the propagation of light in an expanding universe [1285] and the impact ofextreme values of the cosmological constant on the physics of BH collisions [1286].

6. Effective-One-Body and Phenomenological modelsContributor: T. Hinderer

This section surveys the status of two main classes of models that are currentlyused in GW data analysis: (i) the EOB approach, which describes both thedynamics and waveforms in the time domain for generic spinning binaries, and (ii)the phenomenological approach (Phenom), which provides a closed-form descriptionof the waveform in the frequency domain and includes the dominant spin effects.The discussion below reviews mainly the current state-of-the-art models; a morecomprehensive overview of prior work can be found in review articles such as Refs. [1287–1289].

6.1. Effective-one-body models

The EOB approach was introduced in [1290, 1291] as a method to combineinformation from the test-particle limit with PN results (for early ideas in this spirit,see Ref.[1292]). The model comprises a Hamiltonian for the inspiral dynamics, aprescription for computing the GWs and corresponding radiation reaction forces, anda smooth transition to the ringdown signals from a perturbed final BH. The idea isto map the conservative dynamics of the relative motion of two bodies, with massesm1,2, spins S1,2, orbital separation x and relative momentum p, onto an auxiliaryHamiltonian description of an effective particle of mass µ = m1m2/(m1 + m2) andeffective spin S∗(S1,S2,x,p) moving in an effective spacetime geff

µν(M,SKerr; ν) that ischaracterized by the total mass M = m1 + m2, symmetric mass ratio ν = µ/M , andtotal spin SKerr(S1,S2). The basic requirements on this mapping are that (i) the test-particle limit reduces to a particle in Kerr spacetime, and (ii) in the weak-field, slow-motion limit the EOB Hamiltonian reduces to the PN Hamiltonian, up to canonicaltransformations. These considerations, together with insights from results in quantum-electrodynamics [1293], scattering theory [1008, 1036], and explicit comparisons of theaction variables [1290] all lead to the “energy map” given by

H = M

√√√√1 + 2ν(Heff

µ− 1

), (11)


where H and Heff are the Hamiltonians describing the physical binary system and theeffective particle respectively.

The details of the effective Hamiltonian Heff are less constrained by theoreticalconsiderations, and different descriptions have been proposed that differ in the structureof the Hamiltonian, the form of the potentials characterizing the effective spacetime, andthe spin mapping between the physical and effective systems. The differences betweenthese choices reflect current limitations of theoretical knowledge; they all agree in thePN and nonspinning test-particle limit. For the structure of Heff , the incarnation ofthe EOB model of Refs. [1112, 1294–1297] imposes that the limiting case of a spinningtest-particle in Kerr spacetime must be recovered, to linear order in the test-spin. Theversion of the model of Refs. [1298–1302] does not include test-particle spin effects, whichenables a more compact description. These different choices also result in different spinmappings, for both the Kerr parameter and the spin of the effective particle. Finally,these two branches of EOB models also employ different ways to re-write the potentialsthat are calculated as a Taylor series in a PN expansion in a “re-summed” form. Thismeans that an empirically motivated non-analytic representation is used that consistseither of a Pade-resummation [1298–1300] or has a logarithmic form [1295, 1296]. Inaddition to the above choices for describing the strong-field, comparable-mass regime,the models also include parameterized terms whose coefficients are functions of themass and spin parameters that are fixed by comparisons to NR results. In the model ofRef. [1112, 1295–1297], these calibration parameters are constrained by the requirementthat the model must reproduce the GSF results for the ISCO shift in Schwarzschild [817].

The radiative sector in the EOB model is described by so-called factorizedwaveforms (instead of a PN Taylor series expansion) that are motivated from thestructure of waveforms in the test-particle limit and have the form [1303, 1304]

hinsp−plunge`m (t) = h

(N,ε)`m S

(ε)eff T`m e

iδ`m f`mN`m . (12)

Here, h(N,ε)`m is the Newtonian contribution, and S(ε)

eff is a certain effective “source term”that, depending on the parity ε of the mode, is related to either the energy or theangular momentum. The factor T`m contains the leading order logarithms arising fromtail effects, the term eiδ`m is a phase correction due to sub-leading order logarithmsin hereditary contributions, while the function f`m collects the remaining PN terms.Finally, the factor N`m is a phenomenological correction that accounts for deviationsfrom quasi-circular motion [1305] and is calibrated to NR results. The modes fromEq. (12) are used to construct dissipative forces F , in terms of which the equations ofmotion are given by

dx

dt= ∂H

∂p,

dp

dt= −∂H

∂x+ F , (13)

dS1,2

dt= ∂H

∂S1,2× S1,2. (14)

This EOB description of the inspiral-plunge dynamics is smoothly connected to themerger-ringdown signal in the vicinity of the peak in the amplitude |h22|. To perform this


matching, initial nonspinning models and the aligned-spin models SEOBNRv1 [1296]and SEOBNRv2 [1297, 1306] (as well as in the models of the IHES group [1298, 1307])used a superposition of damped sinusoids similar to quasi-normal modes [1291]. Thismethod is inspired by results for the infall of a test-particle into a BH and the close-limitapproximation [1308]. More recently, a simpler phenomenological fit of the amplitudeand phase inspired by the rotating source approximation [1309], which was adapted tothe EOB context in [1310], has become standard in SEOBNRv4 [1112]. This methodprovides a more stable and controlled way to connect the inspiral-plunge to the ringdown.A further key input into the merger-ringdown model is the frequency of the least-dampedquadrupolar quasinormal mode σ220 of the remnant based on a fitting formula from NRfor the mass and spin of the final object given the initial parameters, with the currentlymost up-to-date fit from [1112].

Generic spin precession effects are also included in the model, by starting froma calibrated spin-aligned model and transforming to the precessing frame as dictatedby the precession equations derived from the EOB Hamiltonian [1311, 1312], withoutfurther calibrations. The most recent refinement for generic precessing binaries fromRef. [1312] is known as SEOBNRv3.

For non-vacuum binaries involving e.g. neutron stars, exotic compact objects, orcondensates of fundamental fields around BHs, several effects of matter change the GWsfrom the inspiral relative to those from a BH binary, as described in Chapter III, Sec. 4.4,where also the signatures from such systems generated during the merger and ringdownare covered. Effects during the inspiral include spin-induced deformations of the objects,tidal effects, the presence of a surface instead of an event horizon, tidal excitation ofinternal oscillation modes of the objects, and more complex spin-tidal couplings. Twoclasses of EOB models for such systems are currently available, corresponding to the twodifferent baseline EOB models for BHs. One is known as TEOBRESUMS [1313] andincorporates the effects of rotational deformation and adiabatic tidal effects [1314, 1315]in re-summed form that was inspired by [830] and augmented as described in [442]. Theother model is known as SEOBNRv4T and includes the spin-induced quadrupole as wellas dynamical tides from the objects’ fundamental oscillation modes [450, 1316].

6.2. Phenomenological (Phenom) models

The aim of the Phenom models is to provide a simple, efficient, closed-formexpression for the GW signal in the frequency domain [1317] by assuming the schematicform

hphen(f ; ~α; ~β) := A(f ; ~α)eiΨ(f ;~β), (15)

where ~α and ~β are amplitude and phase parameters in the model. Phenomenological(“Phenom”) models were first developed for nonspinning binaries in Refs. [1317, 1318]and subsequently refined to include aligned spins [1319] known as “PhenomB”. Thismodel employed only a single weighted combination of the individual BH spinscharacterizing the dominant spin effect in the GWs, and was further refined in [1320]


(“PhenomC”), and in [1101, 1111] (“PhenomD”). The latter has been calibrated usingNR data for mass ratios up to 1:18 and dimensionless spins up to 0.85 (with a larger spinrange for equal masses). An effective description of the dominant precession effects hasalso been developed [1115] (“PhenomP”) [1321, 1322]. The PhenomP model providesan approximate mapping for obtaining a precessing waveform from any non-precessingmodel, with PhenomD being currently used as the baseline.

To construct these state-of-the-art Phenom models, the GW signal is divided intothree main regimes: an inspiral, intermediate region, and merger-ringdown. The ansatzfor the inspiral model is the PN result for the frequency-domain phasing obtained fromenergy balance in the stationary phase approximation (“TaylorF2”), accurate to 3.5PNin the nonspinning and linear-in-spin sector, and to 2PN in spin-spin terms. This hasthe form

ΨTF2 = 2πftc − φc − π/4 + 3128ν (πMf)−5/3

7∑i=0

ci(πMf)i/3, (16)

where ci are PN coefficients. The Phenom models add to this auxiliary terms so thatthe inspiral phase becomes

Ψ = ΨTF2 + 1ν

6∑i=0

σifi/3, (17)

where σi are phenomenological coefficient with σ1 = σ2 = 0.A different ansatz is made for the late inspiral and merger-ringdown signals that

are likewise closed-form expressions involving the frequency and phenomenologicalparameters; in total the model involves 17 phenomenological parameters. For the lateinspiral portion, these are mapped to the set of two physical parameters (ν, χPN), whereχPN is defined by

χPN = χeff −38ν113 (χ1 + χ2) , (18)

χeff = m1

M~χ1 · LN + m2

M~χ2 · LN , (19)

where LN is the direction of the Newtonian angular momentum and χi = Si/m2i . χPN

describes the dominant spin effect (the dependence on M is only a rescaling). Themapping is done by assuming a polynomial dependence σi = ∑

bijkνjχkeff to quadratic

order in ν and third order in χPN so that the coefficients vary smoothly across theparameter space. Finally, the coefficients are calibrated to a large set of hybridwaveforms, which themselves are formed using the uncalibrated SEOBNRv2 model.

Spin effects in the Phenom models are described by several different combinationsof parameters. In the aligned-spin baseline model, the description of the early inspiraldepends on both spin parameters χ1, χ2 from PN. In the later inspiral regime, spineffects are described by the effective combination χPN described above, while the merger-ringdown model is expressed in terms of the total spin. For generic spin orientations,an additional parameter χp that characterized the most readily measurable effects ofprecession is included in the model. The defnition of the precessional parameter χp


is motivated by the observation that waveforms are simpler in the coprecessing framethat is aligned with the direction of the dominant radiation emission [1048]; it is givenby [1322]

χp = 1B1m2

1max(B1S1⊥, B2S2⊥), (20)

where B1 = 2 + 3m2/(2m1) and B2 = 2 + 3m1/(2m2) assuming the conventionm1 > m2. Starting from an aligned-spin frequency-domain waveform model, anapproximate precessing waveform is constructed by “twisting up” the non-precessingwaveform with the precessional motion based on a single-spin PN description for theprecession angles [1115, 1321]. While there are some broad similarities of PhenomP withthe way the precessing EOB model is constructed, the two approaches differ in severaldetails, as explained in Sec. IV of Ref. [1312].

For non-BH objects, the effects of rotational and tidal deformations are included inthe phasing using either PN information [1323–1325] or, for the case of binary neutronstars, using a model calibrated to NR [1326, 1327].

6.3. Remaining challenges

Important advances that all EOB and Phenom models aim to address in the near-future are higher modes, parameter space coverage (especially in the mass ratio) andinclusion of all available theoretical knowledge, reduction of systematic errors, and goingbeyond circular orbits.

To date, most of the effort in calibrating the EOB and Phenom models has focusedon the (2, 2) mode, although for the special case of nonspinning binaries an EOBmultipolar approximant is available [1328]. An accurate model of higher modes isimportant for robust data analysis, especially for spinning binaries. Work is ongoing toaddress this within the EOB model [1329] and with studies that will inform the Phenomapproach [1330]. The parameter space over which current models have been calibratedand tested is limited by available NR simulations of sufficient accuracy and length (seeSec. 4), and systematic uncertainties remain a concern.

Extending the range in parameter space ties into the pressing issue that manyavailable results from GSF calculations are not currently incorporated in these models.One of the obstacles is that GSF data for circular orbits only make sense above the “lightring” radius and cannot directly inform the EOB model across and below that radius.This issue is a further subject of ongoing work. Efforts are also underway to includeeffects of eccentricity [1331–1333]. Effects such as the motion of the center-of-mass,BH radiation absorption, and radiation reaction for spins are also not yet included incurrent models.

For EOBmodels, efficiency for data analysis is a further concern, because computingEOB waveforms is relatively time-consuming. To overcome this issue, reduced-ordermodels for EOB waveforms in the frequency domain have been developed for aligned-spin binaries [1112, 1334], with work being underway on the larger parameter space


of precessing binaries. Also ongoing is work on theoretical aspects of spin effects inthe EOB model, aimed at devising an improved description that is more robust andcomputationally efficient.

6.4. Tests of GR with EOB and Phenom models

The effective models described above have been used for tests of GR and exoticobjects, e.g. in [17]. One of the parameterized tests makes use of the general framework“TIGER” [1335] that has been implemented for data analysis, where parameterizedfractional deviations from the GR coefficients are included in the frequency-domainphase evolution from Eq. (16). In recent studies, e.g. [17, 18], this framework wasused on top of the PhenomD inspiral model [1101, 1111] to obtain bounds on non-GReffects, where the tests also included parameterized deviations in the intermediate andmerger-ringdown regime. Work is ongoing to implement tests of GR within the EOBapproach [1336]. Finally, work is also underway to include in these models the option totest for exotic physics manifest in spin-quadrupole and tidal effects during the inspiral.Despite this recent progress on frameworks to test GR and the nature of BHs, substantialfurther work will be necessary to refine the level of sophistication and physical realismof such tests.

7. Source modelling and the data-analysis challengeContributor: J. R. Gair

Models of GW sources form an essential component of data analysis for GWdetectors. Indeed, many of the scientific objectives of current and future GW detectorscannot be realised without having readily available accurate waveform models. In thissection we first provide a brief overview of the primary techniques used for GW dataanalysis, and follow this with a discussion of the additional challenges that data analysisfor future GW detectors will pose and the corresponding requirements that this will placeon waveform models.

7.1. Overview of current data analysis methods

The LIGO-Virgo Collaboration (LVC) uses a variety of techniques to first identifyand then characterise GW transients identified in their data. The majority of these makeuse of signal models in some way. We briefly describe these methods in the following.

7.1.1. Unmodelled searches A number of LVC analysis pipelines are targeted towards“burst” sources for which the waveforms are not well modelled. These includedCoherent Wave Burst [1337], X-pipeline [1338, 1339] and BayesWave [1340]. Allthree algorithms make use of the fact that there are multiple detectors in the LVCnetwork, which allows signals (common between different detectors) to be distinguishedfrom noise (generally uncorrelated between detectors). The algorithms differ in their


implementation. Coherent Wave Burst and X-pipeline operate on spectrograms (time-frequency) maps of the data, computed using wavelet transforms in the first case anda Fourier transform in the second. Both algorithms then identify “hot” pixels thathave particularly large power, carry out clustering to identify candidate events, i.e.,continuous tracks of excess power, in individual detectors, and then impose consistencyin the properties of the tracks identified in the different detectors. BayesWave takes aslightly different approach, using Bayesian techniques to construct a model for the noisein each detector and any signal present. The signal is constructed as a superpositionof wavelets, and reversible jump Markov Chain Monte Carlo is used to add or removecomponents from the signal and noise models.

These model-free searches are powerful tools for source identification. Indeed, thefirst algorithm to find GW150914, the first GW event detected by LIGO, was theCoherent Wave Burst [1, 19], because it was the only online search pipeline runningat the time of the event. However, these algorithms are not as sensitive as searchesbased on models and the second clear GW event, GW151226, was only found with highsignificance by the template-based searches [2]. In addition, parameter estimation canonly be done using models. Moreover, of the three algorithms, only BayesWave is trulyindependent of waveform models. Both Coherent Wave Burst and X-pipeline uses signalinjections in order to determine the optimal choice of the various tunable thresholds inthe algorithms that maximizes distinction between signal and noise. The injections userealistic GW signal waveforms, for which models are needed. BayesWave does not dotuning in this way, although the development of that algorithm and demonstration ofits performance was done using signal injections.

7.1.2. Template based searches The primary search pipelines within the LVC CompactBinary Coalescence (CBC) group use matched filtering. This involves using aprecomputed bank of templates of possible GW signals and computing their overlap,i.e., inner product, with the data. The template bank needs to fully cover the parameterspace so that if a signal is present, at least one of the templates will recover itconfidently. There are two primary matched filtering based searches used by the LVC— pyCBC [1341, 1342] and gstLAL [1343, 1344]. The two searches differ in a number ofways, such as the choice of template bank and placement, and in the various consistencychecks that are used to compare the signals identified in the different detectors and tocompare the post-signal extraction residual to the expected noise distribution. Werefer the interested reader to the previous references for more details. All the BBHsystems identified by LIGO to date were found by both of these pipelines with veryhigh significance, and several of them would not have been identified using template-freemethods only. Matched filtering is only possible if models of GW signals are availablethat have sufficient fidelity to true signals.

7.1.3. Parameter estimation The other primary area where GW signal models areessential is in parameter estimation. Once a potential signal has been identified


by one of the search pipelines described above, the signal is characterized using theseparate LALInference pipeline [1345] (though other parameter inference methods arealso used [1346, 1347]). LALInference constructs a posterior probability distribution forthe source parameters using Bayes theorem. This relies on a model for the likelihoodwhich is taken to be the likelihood of the noise (assumed Gaussian on the short stretchesof data around each signal). The noise is computed as observed data minus signal, and istherefore a function of the signal parameters and requires an accurate model of potentialsignals. LALInference includes two different algorithms — LALInferenceNest [1348]and LALInferenceMCMC [1349]. The first uses nested sampling to determine theposterior and associated model evidence, while the latter uses Markov Chain MonteCarlo techniques based on the Metropolis-Hastings algorithm.

Parameter estimation is essential to extract physical information from identifiedGW events, and the resulting posterior distributions summarise all our informationabout the properties of the source. Any physical effect that we wish to probe using GWobservations must be included in the signal model used in parameter estimation. Therecent NSB event observed by LIGO, GW170817 [21], showed some evidence for tidaleffects in the signal, which highlighted deficiencies in signal models with tides and theneed for further work in that area [1350].

7.2. Challenges posed by future detectors

The next LIGO-Virgo observing run, O3, is scheduled to start in early 2019, andis expected to have a factor of ∼ 2 improvement in sensitivity over the O2 run thatfinished in August 2017. LIGO/Virgo are expected to complete their first science runsat design sensitivity in the early 2020s [14]. Based on the earlier science runs, severaltens of BBH systems and a small number of NSB events [9, 21] are likely to be detectedin O3. These events will be similar to sources previously identified and so the primarychallenge will be computational — the LVC will need the capability to process multipleevents simultaneously, which implies a need for accurate waveform models that are asfast to generate as possible. Further in the future there are plans for third generationground based detectors, such as the Einstein Telescope [33] and Cosmic Explorer [32],and a space-based GW detector, the Laser Interferometer Space Antenna (LISA) [64].These new detectors will observe new types of source which will pose new modellingchallenges.

7.2.1. Third-generation ground-based detectors Planned third-generation ground-based detectors, like the Einstein Telescope (ET) [33] or the Cosmic Explorer [32], aimto improve on advanced detectors in two ways. Firstly, they aim to have an increasein sensitivity by an order of magnitude. Secondly, they aim to improve low-frequencysensitivity, with the ultimate goal of sensitivity as low as 1Hz, compared to ∼ 30Hz forthe LIGO/Virgo detectors in O2, and 10Hz at design sensitivity. For the ET, these aimswill be achieved by increasing the arm-length to 10km, sitting the detector underground


and using cryogenic cooling of the mirrors. ET will also have three arms in a triangularconfiguration, so that it is equivalent to having two detectors at the same site.

The increase in sensitivity will increase the number of sources detected by abouta factor of a thousand. Data analysis must be able to keep pace with such a highsource volume, which means algorithms must run quickly. This places constraints onthe computational cost of waveform models, which is discussed further below. Theimprovement in low frequency sensitivity significantly increases the duration of anygiven signal in band. At leading order, the time to coalescence of a binary scales likef−8/3 [143]. The NSB event GW170817 was in the LIGO band for 40s, starting from30Hz, and generated 3000 waveform cycles [21]. For a detector with low frequency cut-off at 3Hz, the same source would be in band for ∼ 5h and generate ∼ 105 cycles. Thislonger duration places more stringent requirements on waveform models, since fractionalerrors in the waveforms will need to be small enough that the templates are accurateto within 1 cycle of the 105 in band. This is mitigated partially by the fact that mostof the additional cycles are generated in the weak field where analytic models are wellunderstood.

Third-generation detectors also offer the prospect of detection of new classes ofsources. These include higher-mass BH systems, made possible by the improved low-frequency sensitivity [33], and possibly intermediate mass ratio inspirals (the inspiralsof stellar origin BHs into intermediate mass BHs with mass ∼ 100M) [1351, 1352].The former do not pose additional modelling challenges, as these signals will be wellrepresented by rescaling templates for lower mass systems. The latter, however, liein a regime where both finite mass-ratio effects and higher-order PN effects becomeimportant. The latter require numerical techniques, but these are limited in termsof the number of cycles that can be modelled, while the former requires perturbativetechniques, but are then limited by the size of mass ratio corrections. Any IMRIsobserved are likely to be at very low SNR and so will only be identifiable in the dataif accurate templates are available. Initial attempts to construct hybrid IMRI modelshave been made [1353–1356], but considerable work is still required.

It is also hoped that it will be possible to test GR to high precision withthird-generation ground-based detectors [33]. This requires development of models inalternative theories. This challenge is common to space-based detectors and is discussedfurther in the next section.

7.2.2. The Laser Interferometer Space Antenna LISA is a space-based GW detectorthat has been selected as the third large mission that ESA will launch in its currentprogramme, with a provisional launch date of 2034. LISA will comprise three satellites,arranged in an approximately equilateral triangular formation with 2.5Mkm longarms and with laser links passing in both directions along each arm. By preciselymeasuring the phase of the outgoing and incoming laser light in each arm LISA can dointerferometry and detect GWs. It will operate at lower frequency than the LIGO/Virgodetectors, in the range 0.1mHz–0.1Hz, with peak sensitivity around a few mHz. The


lower frequency sensitivity means that the typical systems that LISA will observe havehigher mass, M ∼ 104–107M. Such massive BHs (MBHs) are observed to reside in thecentres of lower mass galaxies. LISA is expected to observe mergers of binaries comprisedof two such MBHs, which are expected to occur following mergers between the BH hostgalaxies. MBHs are typically surrounded by clusters of stars, which include BHs similarto those observed by LIGO that were formed as the endpoint of stellar evolution. LISAis also expected to observe the EMRIs of such stellar origin BHs into MBHs. In additionto these MBH sources, LISA will also observe stellar compact binaries in the Milky Way,it could detect some of the stellar origin BH binaries that LIGO will observe and maydetect sources of cosmological origin [64]. The latter sources do not pose particularmodelling challenges, but the BH sources do.

In contrast to the LIGO-Virgo network, there will be only one LISA constellation.While the three LISA arms allow, in principle, the construction of two independentdata streams, there will inevitably be some correlation between noise in these channels.In addition, LISA sources are long-lived, lasting months or years in the data set, andso there will be hundreds of sources overlapping in the data. These properties makeit much more difficult to construct unmodelled source pipelines like those used inLIGO and so LISA will rely even more heavily on having models of potential signalsin order to identify them in the data. Typical MBH binary signals will have SNRin the tens to hundreds, with a few as high as a thousand. This allows much moreprecise estimates of parameters, but places much higher demands on the fidelity ofwaveform models. A template accurate to a few percent is fine for characterising asource that has SNR of a few tens as is typical for LIGO/Virgo, but for an SNR of onethousand, the residual after extracting that source will have SNR in the several tens,biasing parameter estimation and contaminating the extraction of subsequent sources.Templates need to be two orders of magnitude more accurate for use in LISA. Thisaccuracy comes coupled to the need for longer duration templates, as for the third-generation ground based detectors, since the signals are present in the data streamfor months. MBH binaries detected by LISA are additionally expected to have highspins [1357], in contrast to the observed LIGO/Virgo sources which are all consistentwith low or zero spin [3, 11, 18, 20, 1358, 1359]. Finally, MBH binaries are more likelyto show precession. The likely presence of these physical effects in observed signals,coupled with the necessity of model-based searches for LISA places strong requirementson the MBH binary waveform models that must be available by the time LISA flies.

For EMRIs, expected SNRs are in the tens [672, 891], but this SNR is accumulatedover ∼ 105 cycles. This makes unmodelled EMRI searches impossible and imposes therequirement on modelled searches that the EMRI templates match the true signals tobetter than one cycle over 105. In addition, all of these cycles are generated in thestrong-field where accurate modelling of the signals is more challenging. This drives therequirement for GSF EMRI models accurate to second order in the mass ratio describedabove. These models must include the effect of high spin in the central MBH, andeccentricity and inclination of the orbit of the smaller BH [699, 1360]. EMRI models


will also have to be computationally efficient. Naively, to match 105 cycles in a parameterspace with 8 intrinsic (and 6 extrinsic) parameters, requires (105)8 ∼ 1040 templates.This is a crude overestimate, but illustrates the complexity of a template-based EMRIsearch, and the need to be able to generate large numbers of templates in a smallcomputational time. Semi-coherent methods have been proposed [698] that have lessstringent requirements, but still need templates accurate for 103 or more cycles.

Finally, one of LISA’s primary science objectives is to carry out high precisiontests of GR, including both tests of strong-field gravity and tests of the nature ofcompact objects by using EMRIs to probe the gravitational field structure in thevicinity of BHs. Many different tests have been proposed and we refer the readerto [1361] for a comprehensive review. Several methods exist for phenomenological tests,which assess consistency of the observed signal with the predictions of GR. However,understanding the significance of any constraints that LISA places, and interpretingany deviations that are identified requires models for deviations from the predictionsof GR in alternate theories. We require strong-field predictions, most likely relyingon numerical simulations, to compare to the merger signals from MBH binaries, whichwill be observable with high significance. We also require predictions for the sorts ofdeviations that might be present in the inspiral phase of EMRIs. The latter need to beaccurate to a part in 105 and must allow for confusion between gravitational effects andeffects of astrophysical origin, e.g., the presence of matter in the system, perturbationsfrom distant objects, etc.

7.3. Computationally efficient models

As described above, a significant obstacle to data analysis for future detectors iscomputational. In order to analyse a large number of potential signals and search largeparameter spaces, we require models that capture all the main physics reliably but canbe evaluated rapidly. We describe the current status and outstanding challenges here.

7.3.1. Reduced-order models In the context of LIGO/Virgo, interest in developingcomputationally efficient representations of waveform models arose due to the highcost of parameter estimation algorithms [1345]. This led to the development ofreduced-order or surrogate models. The gstLAL search algorithm [1343, 1344] usesa singular-value decomposition (SVD) of the signal and noise parameter spaces toefficiently identify candidate signals. However, for parameter estimation we also needto quickly map such a representation onto physical parameters. One approach is totake Fourier representations of gravitational waveforms, construct an SVD of bothof these and finally construct a fit to the dependence of the SVD coefficients onsource parameters [1334, 1362]. An alternative approach is to find a reduced-orderrepresentation of the waveform parameter space. A set of templates covering thewhole parameter space is constructed and then waveforms are added sequentially tothe reduced-basis set using a greedy algorithm. At each stage the template that is least


well represented by the current reduced basis is added to the set [1363, 1364]. Thisprocedure is terminated once the representation error of the basis is below a specifiedthreshold, and typically the number of templates in the final basis is one or more orders ofmagnitude less than the size of the original set. Representing an arbitrary template withthe basis is then achieved by requiring the basis representation to match the waveform ata specific set of points, i.e., interpolation rather than projection. These points are chosenby another greedy algorithm — picking the points at which the difference between thenext basis waveform and its interpolation on the current basis is biggest. This methodof representation then allows the likelihood of the data to be written as a quadraturerule sum over the values of the desired waveform at the interpolation points [1365]. Thisis typically a far smaller number of points than the original time series and so is muchcheaper to evaluate than the full waveform. For numerical waveforms an additional stepis needed in which the values of the waveform at the desired points is interpolated acrossparameter space [1366].

The first GW application of this reduced-order quadrature method used a simplesine-Gaussian burst model [1365] and the first implementation within the LIGO/Virgoanalysis infrastructure was for a NS waveform model [1367]. Reduced-order models andassociated quadrature rules have subsequently been developed for a number of otherwaveform models, including NR simulations for higher mass BH binaries [1119–1121]and phenomenological inspiral-merger-ringdown waveform models [1368]. Parameterestimation for GW170817 [21] would arguably not have been possible within thetimescale on which that paper was written if the reduced order model waveform hadnot been available.

Some of the existing reduced-order models and reduced-order quadratures will beuseful for analysis of data from future detectors. However, the increased duration ofsignals will typically increase the size of the reduced basis and so work will be needed tooptimize the models. In addition, new types of waveform models including higher spins,lower mass ratios or eccentricities do not yet have available reduced-order models, andso these will have to be developed for application to LISA searches for massive BHs andEMRIs. It is possible that some eventual advanced phenomenological models may ontheir own provide a competitive alternative to Reduced-order models.

7.3.2. Kludges To model EMRIs, “kludge” models have been developed that capturethe main qualitative features of true inspiral signals. There are two main kludgeapproaches. The analytic kludge (AK [706]) starts from GW emission from Keplerianorbits, as described by [143, 1369], and then adds various strong-field effects on top—evolution of the orbit through GW emission, perihelion precession and orbital-planeprecession. The model is semi-analytic and hence cheap to generate, which means ithas been used extensively to scope out LISA data analysis [698, 1370, 1371]. However,it rapidly dephases from true EMRI signals. Versions of the AK that include deviationsfrom GR arising from an excess quadrupole moment [1372] and generic changes inthe spacetime structure [1373] have also been developed. The numerical kludge model


(NK) is built around Kerr geodesic trajectories, with the parameters of the geodesicevolved as the object inspirals, using a combination of PN results and fits to perturbativecomputations [1374]. A waveform is generated from the trajectory by identifying theBoyer-Lindquist coordinates in the Kerr spacetime with spherical polar coordinates andusing a flat-space GW emission formula [1375]. The NK model is quite accurate whencompared to trajectories computed using BH perturbation theory [1375]. Versions ofthe NK model including conservative GSF corrections have also been developed [1376].

Current kludge models are fast but not sufficiently accurate to be used in LISA dataanalysis. The NK model can be improved using fits to improved perturbative resultsas these become available. An osculating element formalism has been developed [1377,1378] that can be used to compute the contribution appropriate corrections to thephase and orbital parameter evolution for an arbitrary perturbing force (see [784] foran example using the GSF). This model is likely to be sufficiently reliable for usein preliminary data analysis, but will have to be continually improved as the GSFcalculations described in Sec. 2 are further developed. Although much faster than GSFmodels, the NK model is likely to be too expensive computationally for the first stagesof data analysis, in which source candidates are identified using a large number oftemplates. The AK model in its current form is potentially fast enough, but is notsufficiently accurate. However, recently a hybrid model called the “Augmented AnalyticKludge” was proposed [881, 1379] that corrects the AK phase and orbital evolutionequations so that they match numerical kludge trajectories. The “Augmented AnalyticKludge” model is almost as fast as the AK model and is almost as accurate as the NKmodel, so it is a promising approach to data analysis, though much work is needed todevelop the preliminary model to include the full range of physical effects.

Computationally efficient models will be essential for identifying sources in datafrom current and future GW detectors. However, they can only be constructed ifaccurate physical waveform models have been developed, and the final stage of anyparameter estimation pipeline must use the most accurate physical model available inorder to best extract the source parameters. Accurate source modelling underpins allGW data analysis.

8. The view from Mathematical GRContributor: Piotr T. Chruściel

The detection of gravitational waves presents formidable challenges to themathematical relativist: can one invoke mathematical GR to provide an unambiguousinterpretation of observations, and a rigorous underpinning for some of theinterpretations already made? Here we briefly discuss some of the issues arising, focusingon questions that are tractable using available methods in mathematical GR. We shallsidestep a host of other important problems, such as that of the global well-posedness ofthe Cauchy problem for binary BHs, or the related problem of cosmic censorship, whichare currently completely out of reach to mathematical GR.


8.1. Quasi-Kerr remnants?

A working axiom in the GW community is that Kerr BHs exhaust the collection ofBH end states. There is strong evidence for this, based on the “no-hair” theorem,which asserts that suitably regular stationary, analytic, non-degenerate, connected,asymptotically flat vacuum BHs belong to the Kerr family; see Ref. [1380] for precisedefinitions and a list of many contributors (see also Secs. 4.1 and 4.2 of Chapter IIIfor a discussion on no-hair theorems in the context of beyond-GR phenomenology).The analyticity assumption is highly undesirable, being rather curious: it implies thatknowledge of a metric in a small space-time region determines the metric throughoutthe universe. This assumption was relaxed in [1381] for near-Kerr configurations andin [1382] for slowly rotating horizons, but the general case remains open. Next, thenon-degeneracy assumption is essentially that of non-vanishing surface gravity: themaximally spinning Kerr BHs are degenerate in this sense. One often discards theextremal-Kerr case by declaring that it is unstable. This might well be the case, butit is perplexing that the spin range of observed SMBH candidates is clearly biasedtowards high spin, with a possible peak at extremality [263, 1383]. (Note, however, thatthese studies have an a priori assumption built-in, that the BHs are Kerr.) Finally,while the connectedness assumption is irrelevant when talking about an isolated BH,it is quite unsatisfactory to have it around: what mechanism could keep a many-BHvacuum configuration in a stationary state? All this makes it clear that removingthe undesirable assumptions of analyticity, non-degeneracy, and connectedness in theuniqueness theorems should be a high priority to mathematically-minded relativists.

8.2. Quasi-normal ringing?

The current paradigm is that, after a merger, the final BH settles to a stationary oneby emitting radiation which, at the initial stage, has a characteristic profile determinedby quasi-normal modes. This characteristic radiation field provides a very useful toolfor determining the final mass and angular momentum of the solution. (Note that theextraction of these parameters from the ringdown profile assumes the final state to beKerr, taking us back to the issues raised in the previous paragraph...) All this seemsto be well supported by numerics. But we are still far behind by way of mathematicalproof. Work by Dyatlov [1384, 1385] has rigorously established that quasi-normal modesare part of an asymptotic expansion of the time-behaviour of linear waves on the Kerr-deSitter background with a nonzero cosmological constant. The asymptotically flat Kerrcase turns out to be much more difficult to tackle, and so far no rigorous mathematicalstatements are available for it in the literature. The whole problem becomes evenmore difficult when non-linearities are introduced, with no results available so far. Therecently announced proofs of nonlinear stability of the Schwarzschild solution withinspecific families of initial data [1386, 1387] might serve as a starting point for furtherstudies of this important problem.


8.3. Quasi-local masses, momenta and angular momenta?

It was mind-blowing, and still is, when LIGO detected two BHs of about 35 and30 solar masses merging into one, of about 60 solar masses. But the question arises,what do these numbers mean? Assuming the end state to be Kerr, the last part ofthe statement is clear. But how can the pre-merger masses be determined, or evendefined, given the non-stationarity of the BBH configuration? Each NR group usesits own method for assigning mass and spin parameters to the initial datasets used intheir simulations, and it is not a priori clear how these numbers relate to each other.EOB models likewise employ their own definitions of mass and spin. Any differencesare most likely irrelevant at the current level of detection accuracy. But one hopes thatit will become observationally important at some stage, and is a fundamental issue inour formulation of the problem in any case.

To address the issue requires choosing benchmark definitions of quasi-local mass,momentum, and angular momentum that can be blindly calculated on numerical initialdatasets, without knowing whether the data were obtained, e.g., by solving the spacelikeconstraints with an Ansatz, or by evolving some other Ansatz, or by matching spacelikeand characteristic data. One can imagine a strategy whereby one first locates theapparent horizons within a unique preferred time-slicing (e.g., maximal, keeping in mindthat apparent horizons are slicing-dependent), and then calculates the chosen quasi-localquantities for those. There exists a long list of candidates for quasi-local mass, withvarious degrees of computational difficulty, which could be used after extensive testing,all to be found in [1388]; alphabetically: Bartnik’s, Brown-York’s, Hawking’s (directlyreadable from the area), Kijowski’s, Penrose’s, with the currently-most-sophisticatedone due to Chen, Wang and Yau (see [1389] and references therein). The issue is notwhether there is an optimal definition of quasi-local quantities, which is an interestingtheoretical question on its own, unlikely to find a universally agreed answer, but whetherthere is a well-defined and computationally convenient one that can be used for scientificcommunication. Such a definition would need to have an unambiguous Newtonian limit,necessary for making contact with non-GR astrophysical observations. Incidentally, inrecent work [1390], the Wang and Yau mass has been calculated for balls of fixed radiusreceding to infinity along null geodesics. The key new observation is that the leadingorder volume integrand is the square of the Bondi news function. This is accompanied byan integral over the surface of the bounding spheres which deserves further investigation,numerically or otherwise.

8.4. Quasi-mathematical numerical relativity?

To an outsider, NR looks like a heroic struggle with a quasi-impossible task, which,after years of inspired attempts, resulted in a maze of incredibly complicated codes thatmanage to perform the calculations related to the problem at hand. It is conceivablethat there is no way to control the whole construction in a coherent way. However,some mathematically minded outsiders would like to get convinced that the numerical


calculations are doing what they are supposed to do; compare [1391, 1392]. In otherwords, is it possible to show that, at least some, if not most, if not all of the currentnumerical approximations to Einstein equations would converge to a real solution of theproblem at hand if the numerical accuracy could be increased without limit? Standardconvergence tests, or checks that the constraints are preserved, are of course veryimportant, but they cannot on their own settle the point. A more rigorous proof ofconvergence is desirable.


Chapter III: Gravitational waves and fundamental physics

Editor: Thomas P. Sotiriou

1. Introduction

Gravity is arguably the most poorly understood fundamental interaction. It isclear that General Relativity (GR) is not sufficient for describing the final stages ofgravitational collapse or the very early universe. Additionally, a deeper understandingof the role of gravitation seems to be a necessary ingredient for solving almost anyother major challenge in fundamental physics, cosmology, and astrophysics, such as thehierarchy problem, or the dark matter (DM) and dark energy problems. Gravitationalwaves (GWs) promise to turn gravity research into a data-driven field and potentiallyprovide much needed insights. However, one needs to overcome a major obstacle first:that of extracting useful information about fundamental physics from GW observations.The reason this is a challenge should have become apparent in the previous chapter.The signal is buried inside noise and extracting it requires precise modelling. Doingso in GR is already a major feat and it only gets harder when one tries to add newingredients to the problem. Nonetheless there is very strong motivation. Black holebinary (BBH) mergers are among the most violent events in the universe and some ofthe most interesting and exotic phenomena are expected to take place in the vicinity ofBHs.

This chapter focusses on how BHs can be used to probe fundamental physicsthrough GWs. The next section sets the background by discussing some examples ofbeyond-GR scenarios. Sec. 3 summarises the techniques that are being used or developedin order to probe new fundamental physics through GWs. Sec. 4 is devoted to the effortsto probe the nature and structure of the compact objects that are involved in binarymergers. Finally, Sec. 5 provide a thorough discussion on what GW observation mightreveal on the nature of DM.

2. Beyond GR and the standard model

2.1. Alternative theories as an interface between new physics and observations

The most straightforward way to test new fundamental physics with GWs fromBBH coalescences would be the following: select one’s favourite scenario and model thesystem in full detail, extract a waveform (or better a template bank), and then look forthe prediction in the data. The technical difficulties of doing so will become apparentbelow, where it will also be apparent that the required tools are at best incomplete andrequire further development. However, there is a clear non-technical drawback in thisapproach as well: it assumes that one knows what new fundamental physics to expectand how to model it to the required precision. One should contrast this with the fact


that most quantum gravity candidates, for example, are not developed enough to giveunique and precise predictions for the dynamics of a binary system. Moreover, ideallyone would want to obtain the maximal amount of information from the data, instead ofjust looking for very specific predictions, in order not to miss unexpected new physics.Hence, the question one needs to ask is what is the optimal way to extract new physicsfrom the data?

GW observations test gravity itself and the way matter interacts through gravity.Hence, at the theoretical level, they can test GR and the way GR couples to the StandardModel (SM) and its extensions. This suggests clearly that the new fundamentalphysics that can leave an imprint on GW observations can most likely be effectivelymodelled using an alternative theory of gravity. Recall that alternative theories ofgravity generically contain extra degrees of freedom that can be nonminimally coupledto gravity.

The advantages of this approach are: (i) Alternative theories of gravity can actas effective field theories for describing certain effects and phenomena (e.g. violationsof Lorentz symmetry or parity) or be eventually linked to a specific more fundamentaltheory; (ii) they can provide a complete framework for obtaining predictions for binaryevolutions and waveforms, as they come with fully nonlinear field equations; (iii) Theirrange of validity is broad, so they allow one to combine constraints coming fromthe strong gravity regime with many other bounds from e.g. the weak field regime,cosmology, astrophysics, laboratory tests, etc. The major drawback of this approach isthat it requires theory-dependent modelling, which can be tedious and requires one tofocus on specific alternative theories of gravity.

In this Section we will give a brief overview of some alternative theories that can beused to model new physics in GW observations. This is not meant to be a comprehensivelist nor is it our intent to pinpoint interesting candidates. We simply focus on theoriesthat have received significant attention in the literature in what regards their propertiesin the strong gravity regime and to which we plan to refer to in the coming Sections.

It is worth mentioning that a complementary approach is to use theory-agnosticstrong field parametrisations. Their clear advantage is that they simplify the modellingdrastically and they render it theory-independent. However, they fall short in points(i)-(ii) above. In specific, it is no always straightforward to physical interpret them,they typically describe only part of the waveform, and they do not allow one to combineconstraints with other observations unless interpreted in the framework of a theory.Strong field parametrisations and their advantanges and disadvantages will be discussedin Secs. 3.1, 3.2 and 3.3.

2.2. Scalar-tensor theoriesContributors: T. P. Sotiriou, K. Yagi

One of the simplest ways to modify GR is to introduce a scalar field that isnonminimally coupled to gravity. In many cases, the term scalar-tensor theories refers


to theories described by the action

Sst = 116π

∫d4x√−g(ϕR− ω(ϕ)

ϕ∇µϕ∇µϕ− V (ϕ)

)+ Sm(gµν , ψ) , (21)

where g is the determinant of the spacetime metric gµν , R is the Ricci scalar of themetric, and Sm is the matter action. ψ is used to collectively denote the matter fields,which are coupled minimally to gµν . The functions ω(ϕ) and V (ϕ) need to be specifiedto identify a specific theory within the class. Brans-Dicke theory is a special case withω = ω0 =constant and V = 0 [1199].

It is common in the literature to rewrite the action in terms of a different setof variables. In particular, the conformal transformation gµν = Gϕgµν , where G isNewton’s constant, and the scalar field redefinition 2ϕdφ =

√2ω(ϕ) + 3 dϕ, can be

employed to bring the action (21) to the form

Sst = 116πG

∫d4x

√−g(R− 2gνµ∂νφ∂µφ− U(φ)

)+ Sm(gµν , ψ) , (22)

where U(φ) = V (ϕ)/(Gϕ)2. The set of variables (gµν , φ) is known as the Einstein frame,whereas the original set of variables (gµν , ϕ) is known as the Jordan frame. Quantitieswith a hat are defined with gµν . In the Einstein frame, scalar-tensor theories seeminglytake the form of Einstein’s theory with a minimally coupled scalar field (hence thename). However, this comes at a price: φ now couples to the matter field ψ, as canbe seen if Sm is written in terms of gµν , φ, and ψ. Hence, if one sticks to the Einsteinframe variables, one would infer that matter experiences an interaction with φ (fifthforce), and hence it cannot follow geodesics of gµν . In the Jordan frame instead, thereis no interaction between ϕ and ψ (by definition), and this implies that matter followsgeodesics of gµν . This line of thinking allows one to ascribe a specific physical meaningto the Jordan frame metric, but it does not change the fact that the two frames areequivalent descriptions of the same theory (see Ref. [1393] and references therein for amore detailed discussion).

Scalar-tensor theories are well-studied but generically disfavoured by weak fieldobservations, e.g. [1207, 1394]. In brief, weak field tests dictate that the scalar field, if itexists, should be in a trivial configuration around the Sun and around weakly gravitatingobjects, such as test masses in laboratory experiments. These precision tests translateto very strong constraints on scalar-tensor theories; strong enough to make it unlikelythat there can be any interesting phenomenology in the strong gravity regime, which isour focus here. There is one notable known exception [1205]. To introduce these modelsit is best to first set V = U = 0 and then consider the equation of motion for φ, asobtained from varying action (22). It reads

φ = −4πGα(φ)T , (23)

where T ≡ T µµ, Tµν ≡ −2(−g)−1/2δSm/δgµν is the Einstein frame stress energy tensor,

and

α(φ) = −[2ω(ϕ) + 3]−1/2 . (24)


Solution with φ = φ0 = constant are admissible only if α(φ0) = 0. This conditionidentifies a special class of theories and translates to ω(ϕ0) → ∞ in the Jordan frame,where ϕ0 is the corresponding value of ϕ. For φ = φ0 solutions the correspondingmetric is identical to the GR solution for the same system. However, these solutionsare not unique. It turns out that they are preferable for objects that are less compactthan a certain threshold, controlled by the value of α′(φ0). When compactness exceedsthe threshold, a nontrivial scalar configuration is preferable, leading to deviations fromGR [1201, 1205, 1395]. The phenomenon is called spontaneous scalarization and it isknown to happen for neutron stars (NSs).

A NSB endowed with non-trivial scalar monopolar configurations would loseenergy through dipole emission [1396] and this would affect the orbital dynamics.More specifically, this would be an additional, lower-order, contribution to the usualquadrupolar GW emission that changes the period of binary pulsars. Hence, binarypulsar constraints are extremely efficient in constraining spontaneous scalarization andthe original scenario is almost ruled out [1397, 1398], with the exception of rapidlyrotating stars. However, dipolar emission can be avoided if the scalar-scalar interactionbetween the stars is suppressed. This is the case is the scalar is massive [1213, 1223].The mass defines a characteristic distance, beyond which the fall-off of the scalar profileis exponential (as opposed to 1/r), so if the value of the mass is in the right range,the stars can be scalarized and yet the binary will not exhibit any appreciable dipolaremission above a given separation. One can then avoid binary pulsar constraints and stillhope to see some effect in GW emission from the late inspiral, merger, and ringdown.There is also the possibility that NSs will dynamically scalarize once they come closeenough to each other [1226, 1227, 1399–1401].

It should be noted that the known scalarization scenario faces the followingissue. Usually, calculations assume flat asymptotics with a vanishing gradient for thescalar. However, realistically the value of the scalar far away from the star changes atcosmological timescales and it turns out that this change is sufficient to create conflictwith observations [1402, 1403]. This problem can be pushed into the future by tuninginitial data, so it can technically be easily avoided. Nonetheless, one would eventuallyneed to do away with the tuning, presumably by improving the model.

Black holes in scalar-tensor theories have received considerable attention, mostnotably in the context of no-hair theorems (e.g. [1404–1406]) and their evasions (seealso Ref. [1407] for a discussion). Sections 4.1 and 4.2 are entirely devoted to this topic,so we will not discuss it further here.

The action of eq. (21) is the most general one can write that is quadratic in thefirst derivatives of the scalar, and hence it presents itself as a sensible effective fieldtheory for a scalar field nonminimally coupled to the metric. However, this actioncan be generalized further if one is willing to include more derivatives. Imposing therequirement that variation leads to field equations that are second order in derivatives


of both the metric and the scalar leads to the action

SH =∫d4x√−g (L2 + L3 + L4 + L5) , (25)

where

L2 = K(φ,X), (26)L3 = −G3(φ,X)φ, (27)L4 = G4(φ,X)R +G4X

[(φ)2 − (∇µ∇νφ)2

], (28)

L5 = G5(φ,X)Gµν∇µ∇νφ

− G5X

6[(φ)3 − 3φ(∇µ∇νφ)2 + 2(∇µ∇νφ)3

], (29)

where X ≡ −12∇

µφ∇µφ, Gi need to be specified to pin down a model within the class,and GiX ≡ ∂Gi/∂X. Horndeski was the first to write down this action in an equivalentform [1408] and it was rediscovered fairly recently in Ref. [1409]. Theories describedby this action are referred to as generalized scalar-tensor theories, Horndeski theories,generalized galileons, or simply scalar-tensor theories. We will reserve the last term herefor the action (21) in order to avoid confusion. The term generalized galileons comesfrom the fact that these theories were (re)discovered as curved space generalization ofa class of flat space scalar theories that are symmetric under φ → φ + cµx

µ + c, wherecµ is a constant one-form and c is a constant (Galilean symmetry) [1410]. In fact, thenumbering of the Li terms in the Lagrangian is a remnant of the original Galileons,where the i index indicates the number of copies of the field. This is no longer true inaction (25), but now the Li term contains i− 2 second derivatives of the scalar.

In what regards strong gravity phenomenology, a lot of attention has been given tothe shift-symmetric version of action (25). If Gi = Gi(X), i.e. ∂Gi/∂φ = 0, then (25) isinvariant under φ → φ+constant. This symmetry protects the scalar from acquiring amass from quantum corrections. It has been shown in Ref. [1411] that asymptoticallyflat, static, spherically symmetric BHs in shift-symmetric Horndeski theories have trivial(constant) scalar configuration and are hence described by the Schwarzschild solution.This proof can be extended to slowly-rotating BHs, in which case the solution is theslowly rotating limit of the Kerr spacetime [1412]. However, there is a specific, uniqueterm in (25) that circumvents the no-hair theorem and leads to BHs that differ fromthose of GR and have nontrivial scalar configurations (hairy BHs) [1412]: φG, whereG ≡ R2 − 4RµνR

µν +RµνρσRµνρσ is the the Gauss-Bonnet invariant.∗ This singles out

SφG = 116πG

∫d4x√−g(R− ∂νφ∂µφ+ αφG

)+ Sm(gµν , ψ) , (30)

as the simplest action within the shift-symmetric Horndeski class that has hairy BHs.α is a coupling constant with dimension of a length squared. Hairy BHs in this theory[1414, 1415] are very similar to those found earlier in the (non-shift-symmetric) theorywith exponential coupling [1416, 1417], i.e. eφG. The theory with exponential coupling

∗ Though this term does not seem to be part of action (25), it actually corresponds to the choiceG5 ∝ ln |X| [1413].


is known as Einstein-dilaton Gauss-Bonnet theory and it arises as a low-energy effectivetheory of heterotic strings [1418, 1419].

The subclass of theories described by the action

Sf(φ)G = 116πG

∫d4x√−g(R− 1

2∂νφ∂µφ+ f(φ)G

)+ Sm(gµν , ψ) , (31)

seems to have interesting BH phenomenology in general. This theory admits GRsolutions if f ′(φ0) = 0 for some constant φ0. Assuming this is the case, it has recentlybeen proven in Ref. [1420] that stationary, asymptotically flat BH solutions will beidentical to those of GR, provided f ′′(φ0)G < 0. A similar proof, but restricted tospherical symmetry, can be found in Ref. [1421]. Theories that do not satisfy thef ′′(φ0)G < 0 exhibit an interesting phenomenon [1420, 1422]: BH scalarization, similarto the NS scalarization discussed above. See also Ref. [1423] for a study of hairy BHsolution in this class of theories and Ref. [1424] for a very recent exploration of linearstability.

The same class of theories can exhibit spontaneous scalarization in NSs [1420]. Onthe other hand, in shift-symmetric Horndeski theories it is known that for NSs the scalarconfiguration will be trivial [1425–1427], provided that the φG terms is absent. Even ifit is there, there will be a faster than 1/r fall-off [1034, 1428]. Hence, for shift-symmetrictheories BH binaries are probably the prime strong gravity system of interest.

It should be noted that the speed of GWs in generalized scalar tensor theories candiffer from the speed of light under certain circumstances and this has been used toobtain constraint recently [674–677, 1429, 1430]. These will be discussed in Sec. 3.1.

As mentioned above, action (25) is the most general one that leads to second orderequations upon variation. Nonetheless, it has also been shown that there exist theoriesoutside this class that contain no other degrees of freedom than the metric and the scalarfield [1431–1436]. These models are often referred to as beyond-Horndeski theories.

Last but not least, dynamical Chern-Simons (dCS) gravity [1437, 1438] is ascalar-tensor theory that introduces gravitational parity violation at the level ofthe field equations. It draws motivation from the standard model [1439], heteroticsuperstring theory [1440], loop quantum gravity [1441–1445] and effective field theoriesfor inflation [1446]. dCS gravity is not a member of the Horndeski class. The scalar isactually a pseudo-scalar, i.e. it changes sign under a parity transformation. Moreover,it couples to the Pontryagin density ∗Rα γδ

β Rβαγδ, where Rα

βγδ is the Riemann tensor,∗Rα γδ

β ≡ 12εγδµνRα

βµν and εγδµν is the Levi-Civita tensor. This coupling term gives riseto third-order derivatives in the field equations. The fact that the theory has higher-order equations implies that it is unlikely it is actually predictive as it stands [1110] (seealso [1447]). Hence, most work has used a specific approach to circumvent that problemthat is inspired by effective field theory and relies on allowing the coupling term tointroduce only perturbative correction. This issue will be discussed in some detail inSec. 3.4.

dCS gravity has received a lot of attention in the strong gravity regime. Non-rotating BH and NS solutions are the same as in GR, as such solutions do not


break parity. On the other hand, slowly-rotating [1448–1450], rapidly-rotating [1451]and extremal [1452, 1453] BHs and slowly-rotating NSs [1035] all differ from thecorresponding GR solutions. In particular, such rotating compact objects acquire ascalar dipole hair and a correction to the quadrupole moment, which modify gravitationalwaveforms from compact binary inspirals [1454–1457] and mergers [1108]. FutureGW observations will allow us to place bounds on the theory that are orders ofmagnitude stronger than the current bounds from Solar System [1458] and table-topexperiments [1450]. Another interesting phenomenon in dCS gravity is gravitationalamplitude birefringence, where the right-handed (left-handed) circularly-polarized GWsare either enhanced or suppressed (suppressed or enhanced) during their propagationfrom a source to Earth, which can be used to probe the evolution of the pseudo-scalar field [1459–1461]. Such GW observations are complementary [1462] to SolarSystem [1463] and binary pulsar [1464, 1465] probes.

For a general introduction to standard scalar-tensor theories see Refs. [1198, 1466]and for generalized scalar-tensor theories see Ref. [1467], and for a recent brief reviewon gravity and scalar fields see Ref. [1468].

2.3. Lorentz violationsContributor: T. P. Sotiriou

Lorentz symmetry is a fundamental symmetry for the standard model of particlephysics, hence it is important to question whether it plays an equally important role ingravity. To address this question one needs to study Lorentz-violating (LV) theories ofgravity for two distinct reasons: (i) to quantify how much one is allowed to deviatedfrom GR without contradicting combined observations that are compatible with Lorentzsymmetry one needs to model the deviations in the framework of a consistent theory;(ii) studying the properties of such theories can provide theoretical insights. We willbriefly review two examples of LV theories here.

Einstein-aether theory (æ-theory) [1469] is described by the action

Sae = 116πG

∫d4x√−g(−R−Mαβ

µν∇αuµ∇βu

ν) (32)

where

Mαβµν = c1g

αβgµν + c2δαµδ

βν + c3δ

αν δ

βµ + c4u

αuβgµν , (33)

ci are dimensionless coupling constants and the aether uµ is forced to satisfy theconstraint u2 ≡ uµuµ = 1. This constraint can be enforced by adding to the actionthe Lagrange multiplier term λ(u2 − 1) or by restricting the variation of the aetherso as to respect the constraint. Due to the constraint, the aether has to be timelike,thereby defining a preferred threading of spacetime by timelike curves, exactly like anobserver would. It cannot vanish in any configuration, including flat space. Hence, localLorentz symmetry, and more precisely boost invariance, is violated. (32) is the mostgeneral action that is quadratic in the first derivatives of a vector field that couples toGR and satisfies the unit constraint. Indeed, æ-theory is a good effective field theory


for Lorentz-symmetry breaking with this field content [1470]. It should be emphasisedthat the aether brings 3 extra degrees of freedom into the theory, 2 vector modes anda scalar (longitudinal) mode. The speed of any mode, including the usual spin-2 modethat corresponds to GWs, can differ from the speed of light and is controlled by a certaincombination of the ci parameters.

One can straightforwardly obtain an LV theory with a smaller field content fromaction (32) by restricting the aether to be hypersurface orthogonal [1471]. This amountsto imposing

uµ = ∂µT√gλν∂λT∂νT

(34)

where T is a scalar and the condition is imposed before the variation (i.e. one varieswith respect to T ). This condition incorporates already the unit constraint u2 = 1 andremoves the vector modes from the theory. It can be used to rewrite the action in termsof the metric and T only. T cannot vanish in any regular configuration that solves thefield equations and its gradient will always be timelike. Hence, the aether is alwaysorthogonal to some spacelike hypersurfaces over which T is constant. That is, T definesa preferred foliation and acts as a preferred time coordinate. Note that it appears inthe action only through uµ and the latter is invariant under T → f(T ), so this is asymmetry of the theory and the preferred foliation can be relabelled freely.

One can actually use the freedom to choose the time coordinate and write thistheory directly in the preferred foliation [1471]. Indeed, if T is selected as the timecoordinate, the T = constant surfaces define a foliation by spacelike hypersurfaces withextrinsic curvature Kij and N and N i are the lapse function and shift vector of thisfoliation. Then uµ = Nδ0

µ, and action (32) takes the form

ST = 116πG′

∫dTd3xN

√h(KijK

ij − λK2 + ξ(3)R + ηaiai), (35)

where ai ≡ ∂i lnN andG′

G= ξ = 1

1− (c1 + c3) , λ = 1 + c2

1− (c1 + c3) , η = c1 + c4

1− (c1 + c3) . (36)

Since T was chosen as a time coordinate, action (35) is no longer invariant under generaldiffeomorphisms. It is, however, invariant under the subset of diffeomorphisms thatpreserve the folation, or else the transformations T → T ′ = f(T ) and xi → x′i =x′i(T, xi), which are now the defining symmetry of the theory from a EFT perspective.In fact, this action can be thought of as the infrared (low-energy) limit of a largertheory with the same symmetry, called Hořava gravity [1472]. The most general actionof Hořava gravity is [1473]

SH = 116πGH

∫dTd3xN

√h

(L2 + 1

M2?

L4 + 1M4

?

L6

), (37)

where L2 is precisely the Lagrangian of action (35). L4 and L6 collectively denote all theoperators that are compatible with this symmetry and contain 4 and 6 spatial derivativesrespectively (in the T foliation). The existence of higher order spatial derivatives implies


that perturbation will satisfy higher order dispersion relations andM? is a characteristicenergy scale that suppresses the corresponding terms. Hořava gravity has been arguedto be power-counting renormalizable and hence a quantum gravity candidate [1472].The basic principle is that the higher-order spatial derivative operators modify thebehaviour of the propagator in the ultraviolet and remove/regulate divergences (seeRefs. [1474, 1475] for a discussion in the context of a simple scalar model). Theprojectable version of the theory [1472, 1476], in which the lapse N is restricted to havevanishing spatial derivatives, has actually been shown to be renormalizable (beyondpower-counting) in 4 dimensions [1477], whereas the 3-dimensional version [1478] isasymptotically [1479]. This restricted version does however suffer from serious viabilityissues at low energies [1472, 1480–1483].

We will not discuss the precise form of L4 and L6 any further, as it not important forlow-energy physics. It should be stressed though that such terms lead to higher ordercorrections to the dispersion relation of GWs and that of the extra scalar excitationHořava gravity propagates. This implies that the speeds of GW polarizations can differfrom the speed of light and acquire a frequency dependence. The relevant constraintsfrom GW observations are discussed in Sec. 3.1 (see also [1484] for a critical discussion).At large wavenumber k, the dispersion relations scale as ω2 ∝ k6 (since by constructionthe leading order ultraviolet operators in the action contain 2 temporal and 6 spatialderivatives) and this implies that an excitation can reach any speed, provided it hasshort enough wavelength. More surprisingly, it has been shown that Hořava gravityexhibits instantaneous propagation even in the infrared limit, described by action(35). In this limit, dispersion relations are linear but, both at perturbative [1485]and nonperturbative level [1486], there exists an extra degree of freedom that satisfiesan elliptic equation on constant preferred time hypersurfaces. This equation is notpreserved by time evolution and hence it transmits information instantaneously withinthese hypersurfaces. For a detailed discussion see Ref. [1487].

It should be clear from the discussion above that both æ-theory and Hořava gravitycan exhibit superluminal propagation and this makes BHs particularly interesting inthese theories. For both theories BHs differ from their GR counterparts and have veryinteresting feature that will be discussed further in Sec. 4.2. For a review see Ref. [1488].

For early reviews on æ-theory and Hořava gravity see Refs. [1489] and [1490],respectively. For the most recent combined constraints see Refs. [1491, 1492] andreferences therein.

2.4. Massive gravity and tensor-tensor theoriesContributor: S. F. Hassan

In the appropriate limit GR reproduces Newton’s famous inverse square law ofgravitation. At the same time, linear excitations in GR (GWs) satisfy a linear dispersionrelation without a mass term. In particle theory terminology, both of these statementsimply that the graviton — the boson that mediates the gravitational interaction — is


massless. Extensions of GR that attempt to give the graviton a mass are commonlyreferred to as massive gravity, bimetric gravity, and multimetric gravity theories. Oneof the original motivations for the study of these theories was the hope that they couldprovide a resolution to the cosmological constant problem by changing the gravitationalinteraction beyond some length scale (controlled by the mass of the graviton). However,a more fundamental motivation emerges from comparing the structure of GR with that ofthe standard model of particle physics that successfully accounts for all known particlesinteractions. This is elaborated below.

The standard model describes particle physics in terms of fields of spin 0 (theHiggs field), spin 1/2 (leptons and quarks) and spin 1 (gauge bosons). At the mostbasic level, these fields are governed by the respective field equation, namely, the Klein-Gordon equation, the Dirac equation and the Maxwell/Yang-Mills/Proca equations.The structures of these equations are uniquely fixed by the spin and mass of the fieldalong with basic symmetry and consistency requirements, in particular, the absence ofghost instabilities, for example, in spin-1 theories. From this point of view, GR is theunique ghost-free theory of a massless spin-2 field; the metric gµν . Then the Einsteinequation, Rµν − 1

2gµνR = 0, with all its intricacies is simply the counterpart for a spin2 field of the massless Klein-Gordon equation φ = 0, or of the Maxwell equation∂µF

µν = 0. However, to describe physics, the standard model contains a second levelof structure: At the fundamental level, all fields appear in multiplets allowing them totransform under symmetries, in particular under gauge symmetries which are in turnresponsible for the renormalizability and hence, for the quantum consistency of thetheory. The observed fields are related to the fundamental ones through spontaneoussymmetry breaking and through mixings. For example, to obtain a single Higgs field,one needs to start with four scalar fields arranged in a complex SU(2) doublet, ratherthan the one scalar field of the final theory. Similarly, although electrodynamics is verywell described by Maxwell’s equations as a standalone theory, its origin in the standardmodel is much more intricate, requiring the spontaneous breaking of a larger gaugesymmetry and a specific mixing of the original gauge fields into the vector potential ofelectromagnetism. Now, one may contrast these intricate features of the standard modelwith GR which is the simplest possible theory of a single massless spin 2 field with noroom for any further structure. The comparison suggests embedding GR in setups withmore than one spin 2 field and investigating the resulting models. The aim would be tosee if the new structures could address some of the issues raised above, either in analogywith the standard model, or through new mechanisms. This will shed light on a cornerof the theory space that has remained unexplored so far.

However, implementing this program is not as straightforward as it may seem. Itwas known since the early 70’s that adding other spin 2 fields to GR generically leads toghost instabilities. The instability was first noted in the context of massive gravity,which contains an interaction potential between the gravitational metric gµν and anondynamical predetermined metric fµν . However it also plagues dynamical theoriesof two or more spin 2 fields. Recent developments since 2010 have shown that one can


indeed construct ghost free theories of massive gravity (with a fixed auxiliary metric)[1493, 1494], or fully dynamical ghost free bimetric theories [1495]. In particular, thebimetric theory is the theory of gravitational metric that interacts with an extra spin-2field in specific ways and contains several parameters. It propagates one massless andone massive spin-2 mode which are combinations of the two metrics. Consequently, itcontains massive gravity and GR as two different limits. For a review of bimetric theorysee [1496].

In the massive gravity limit, as the name suggests, the theory has a single massivespin-2 field that mediates gravitational interactions in the background of a fixed auxiliarymetric. The most relevant parameter then is the graviton mass which is constrainedby different observations (for a review see, [1497]). In particular, consistency with thedetection of GWs by LIGO restricts the graviton mass to m < 7.7 × 10−23eV [11, 17].However, in this limit, it becomes difficult to describe both large scale and small scalephenomena consistently with the same fixed auxiliary metric.

Of course, the bimetric theory is more relevant phenomenologically close to theGR limit, where gravity is described by a mostly massless metric, interacting with amassive spin-2 field. Now the two most relevant parameters are the spin-2 mass m andthe Planck mass M of the second spin-2 field, M MP , where MP is the usual Planckmass of the gravitational metric. The presence of the second parameter M significantlyrelaxes the bounds on the spin-2 mass m. In the most commonly assumed scenarios, thestability of cosmological perturbations in the very early Universe requires M . 100GeV[1498]. In this case, GWs will contain a very small admixture of the massive mode,hence, m is not strongly constrained by LIGO and can be large. A new feature isthat now the massive spin-2 field can be a DM candidate. Depending on the specificassumptions, the relevant allowed mass ranges are estimated as 10−4eV . m . 107eV[1499], or the higher range, 1TeV . m . 100TeV [1500].

While some studies of cosmology in bimetric theory have been carried out, manyimportant features of the theory are yet to be fully explored. One of these is the problemof causality and the well-posedness of the initial value problem. In particular, bigravityprovides a framework for investigating models where the causal metric need not alwayscorrespond to the gravitational metric [1501, 1502]. Another interesting question is if thebimetric interaction potential can be generated through some kind of Higgs mechanism,in analogy with the Proca mass term for spin 1 fields. From a phenomenological pointof view, it is interesting to investigate the massive spin 2 field present in the theory asa DM candidate. Strong field effects in bimetric gravity are expected to show largerdeviations from GR as compared to weak field effects.

While ghost free bimetric models have interesting properties, they do not admitenlarged symmetries, since the two spin 2 fields have only nonderivative interactions.Constructing ghost free theories of spin 2 multiplets with gauge or global symmetries isa more difficult problem which requires constructing ghost free derivative interactions.So far, this is an unresolved problem which needs to be further investigated. If suchtheories exist, they will provide the closest analogues of the standard model in the spin


2 sector.

3. Detecting new fields in the strong gravity regime

As has been discussed in some detail in Section 2, if new physics makes anappearance in the strong gravity regime, it can most likely be modelled as a new fieldthat couples to gravity. In order to detect this field with GW observations, one hasto model its effects on the dynamics of the binary system and produce waveforms thatcan be compared with observations. The 3 distinct phases of a binary coalescence —inspiral, merger, and ringdown — require different modelling techniques. The inspiraland the ringdown are susceptible to different types of perturbative treatment.

During most of the inspiral, the two members of the binary are well separatedand, even though they are strongly gravitating objects individually, they interactweakly with each other. Hence, one can model this interaction perturbatively andthis modeling can be used to produce a partial waveform for the inspiral phase. Itshould be stressed that this gives rise to a perturbation expansion whose parametersdepend on the nature and structure of the members of the binary. Computing theseparameters, known as sensitivities, generally requires modelling the objects composingthe binary in a nonperturbative fashion. Moreover, if the objects are endowed withextra field configurations, then one needs to model them as translating in an ambientfield (in principle created by the companion), in order to faithfully capture the extrainteraction. This is the same calculation one performs to obtain constraints from binarypulsars [1402, 1503]. An alternative is to try to prepare theory-agnostic, parametrisedtemplates. Three different ways to achieve this will be discussed in detail in Sec. 3.2.

The ringdown can be modelled with linear perturbation theory around a knownconfiguration for a compact object, but there are significant challenges. The mostobvious ones are: (i) one needs to first have fully determined the structure of thequiescent object that will be used as background; (ii) the equations describing linearperturbations around such a background are not easy to solve (e.g. separability is anissue); (iii) both the background and the perturbation equations depend crucially onthe scenario one is considering. An in-depth discussion in perturbation theory aroundBHs, and its use to describe the ringdown phase, will be given in Sec. 3.3.

The merger phase is highly nonlinear and modeling it requires full-blown numericalsimulations. A prerequisite for performing such simulations is to have an initial valueformulation and to show that the initial value problem (IVP) is well-posed (in anappropriate sense, as will be discussed in more detail below). This has not beenaddressed for the vast majority of gravity theories that are currently being considered.Sec. 3.4 focuses on the current state of the art in modeling the merger phase. It containsa critical discussion of the IVP in alternative theories of gravity and a summary of knownnumerical results.

Last but not least, one could attempt to detect new fields (polarisations) bythe effect they have on GW propagation, as opposed to generation. This amounts


to parametrizing the dispersion relation of the GWs and constraining the variousparameters that enter this parameterisation, such as the mass, the speed, and theparameters of terms that lead to dispersive effects. The next section is devoted topropagation constraints.

3.1. Gravitational wave propagationContributor: T. P. Sotiriou

According to GR, GWs are massless and propagate at the speed of light, c. Hence,they satisfy the dispersion relation E2 = c2p2, where E is energy and p is momentum.Deviations from GR can introduce modifications to this dispersion relation. LIGO usesthe following parametrisation to model such deviations: E2 = c2p2 + Acαpα, α > 0[1504]∗. When α = 0, A represents a mass term in the dispersion relation. The currentupper bound on the the mass from GW observations is 7.7 × 10−23eV/c2 [11]. Whenα = 2, A represents a correction to the speed of GWs. The fact that GWs from theNSB merger GW170817 almost coincided with the gamma ray burst GRB 170817A hasgiven a spectacular double-sided constraint on the speed of GWs, which should agreewith the speed of light to less than a part in 1015 [21, 23].

All other corrections introduce dispersive effects which accumulate with the distancethe wave had to travel to reach the detector. This enhances the constraints in principle.However, it should be noted that A has dimensions [A] = [E]2−α, which implies thatcorrections that have α > 2 are suppressed by positive powers of cp/M?. M? denotessome characteristic energy scale and it is useful to think of the constraints on A asconstraints on M?. GWs have long wavelengths, so one expects α > 2 corrections to beheavily suppressed unless the aforementioned energy scale is unexpectedly small. Forinstance, for α = 3 the current upper bound on M? is in the 109 GeV range, whereas forα = 4 it drops well below the eV range [11, 1484, 1504, 1506].

It is important to stress that all of the aforementioned constraints need to beinterpreted within the framework of a theory (as is the case for any parametrisation).The mass bound can be turned to a constraint on the mass of the graviton in massivegravity theories, discussed in Sec. 2.4. Interestingly, it is not the strongest constraint onecan obtain [1497, 1507]. All other bounds can in principle be interpreted as constraintson Lorentz symmetry. However, terms with odd powers of α require spatial parityviolations as well and the bounds on terms with α > 2 are probably too weak togive interesting constraints on Lorentz-violating gravity theories [1484]. The double-sided bound on the speed of GWs certainly yields a very strong constraint on Lorentz-violating theories of gravity. However, one has to take into account that such modelsgenerically have a multidimensional parameter space even at low energies and genericallyexhibit extra polarisations, the existence/absence of which can potentially give strongeror additional constraints [1484]. For details on the implications of this bound for Hořavagravity and for Einstein-æther theory, discussed in Sec. 2.3, see Refs. [1491] and [1492]

∗ A more general parametrisation [1505] has been used in Ref. [23].


respectively.Lorentz symmetry breaking is not the only scenario in which GWs propagate at

a different speed than light. Lorentz invariant theories can exhibit such behaviour aswell if they satisfy two conditions: (i) they have extra fields that are in a nontrivialconfiguration between the emission and the detection locations; (ii) these fields coupleto metric perturbations in a way that modifies their propagation. The second conditionsgenerally requires specific types of nonminimal coupling between the extra fields andgravity. It is satisfied by certain generalised scalar tensor theories [1430], discussed inSec. 2.2. Indeed, the bound on the speed has been used to obtain strong constraintson such theories as models of dark energy [674–677, 1429, 1430]. It should be stressed,however, that, due to condition (i), such constraints are inapplicable if one does notrequire that the scalar field be the dominant cosmological contribution that drives thecosmic speed-up. Hence, they should not be interpreted as viability constraints on thetheories themselves, but only on the ability of such models to account for dark energy.

In summary, modifications in the dispersion relation of GWs can be turned intoconstraints on deviations from GR. How strong these constraints are depends on thetheory or scenario one is considering. The clear advantage of propagation constraintsis that they are straightforward to obtain and they do not require modeling of thesources in some alternative scenario. This reveals a hidden caveat: they are intrinsicallyconservative, as they assume that there is no deviation from the GR waveform at thesource. Moreover, as should be clear from the discussion above, propagation constraintshave other intrinsic limitations and can only provide significant constraints for modelsthat lead to deviations in the dispersion relation of GWs for relatively long wavelengths.Hence, one generally expects to get significantly stronger or additional constraints byconsidering the deviation from the GR waveform at the source.

3.2. Inspiral and parametrized templatesContributor: K. Yagi

3.2.1. Post-Newtonian parametrisation Rather than comparing GW data to templatewaveforms in non-GR theories to probe each theory one at a time, sometimes it ismore efficient to prepare parameterized templates that capture deviations away fromGR in a generic way so that one can carry out model-independent tests of gravity withGWs. We will mainly focus on the parameterized modification in the inspiral part ofthe waveform and discuss what needs to be done to construct parameterized waveformsin the merger-ringdown part of the waveform.

One way to prepare such parameterized templates is to treat coefficients of eachpost-Newtonian (PN) term in the waveform phase as an independent coefficient [1508–1510]. For gravitational waveforms of non-spinning BBH mergers in GR, suchcoefficients only depend on the masses of individual BHs. Thus, if GR is correct, any twomeasurements of these coefficients give us the masses, and the measurement of a third


-4 -3 -2 -1 0 1 2 3n PN

10-10

10-8

10-6

10-4

10-2

100

102

104

|β|

GW150914GW151226PSR J0737-3039Solar System

Figure 11. Upper bounds on the ppE parameter β with GW150914 and GW151226entering at different PN orders. For comparison, we also present bounds from binarypulsar observations and Solar System experiments. This figure is taken and editedfrom [1506].

coefficient allows us to carry out a consistency test of GR. This idea is similar to whathas been done with binary pulsar observations with measurements of post-Keplerianparameters [1511, 1512]. Though, it is nontrivial how to perform such tests for spinningBBHs. Moreover, this formalism does not allow us to probe non-GR corrections enteringat negative PN orders due to e.g. scalar dipole emission.

3.2.2. Parameterized post-Einsteinian (ppE) formalism In this formalism the modifiedwaveform in the Fourier domain is given by h(f) = hGR(f) eiβv2n−5 [1513]. Here hGR isthe GR waveform, β is the ppE parameter representing the magnitude of the non-GRdeviation while v is the relative velocity of the binary components. The above correctionenters at nth PN order. Since n is arbitrary, this formalism can capture deviations fromGR that enter at negative PN orders. It has been used in [1506] to probe strong anddynamical field gravity with GW150914 [1] and GW151226 [2]. Bounds on β at each PNorder using a Fisher analysis are shown in Fig. 11. Here, the authors included non-GRcorrections only in the inspiral part of the waveform. Having these bounds on genericparameters at hand, one can map them to bounds on violations in fundamental pillars ofGR. For example, a -4PN deviation maps to time variation of the gravitational constantG, that allows us to check the strong equivalence principle. 0PN (2PN) corrections canbe used to probe violations in Lorentz (parity) invariance.

3.2.3. Phenomenological waveforms The LIGO/Virgo Collaboration (LVC) useda generalized IMRPhenom waveform model [17] by promoting phenomenologicalparameters ~p in the GR waveform to ~p(1 + δp), where δp corresponds to the fractional


deviation from GR. For the inspiral part of the waveform, one can show that thesewaveforms are equivalent to the ppE waveform. LVC’s bounds on non-GR parametersusing a Bayesian analysis on the real data set are consistent with the Fisher bound inFig. 11.

One important future direction to pursue is to come up with a meaningfulparameterization of the waveform in the merger and ringdown phases. The generalizedIMRPhenom waveform model used by LVC does contain non-GR parameters in themerger-ringdown part of the waveform, though it is not clear what they mean physicallyor how one can map such parameters to e.g. coupling constants in each non-GR theory.To achieve the above goal, one needs to carry out as many BBH merger simulations aspossible in theories beyond GR [1107, 1108, 1225, 1514, 1515] to give us some insighton how we should modify the waveform from GR in the merger-ringdown phase.

3.3. Ringdown and black hole perturbations beyond General RelativityContributor: P. Pani

Depending on the physics being tested, the post-merger “ringdown” phase of acompact-binary coalescence might be better suited than the inspiral or the merger.For instance, in the inspiral phase the nature of the binary components shows up onlyat high PN order (though corrections due to extra polarizations can show up at lowPN order) whereas the merger phase requires time-consuming and theory-dependentnumerical simulations. On the other hand, the post-merger phase — during which theremnant BH settles down to a stationary configuration — can be appropriately describedby perturbation theory and it is therefore relatively simple to model beyond GR. If theremnant is a BH in GR, the ringdown phase is well-described by a superposition ofexponentially damped sinusoids,

h+ + ih× ∼∑lmn

Almn(r)e−t/τlmn sin(ωlmnt+ φlmn) , (38)

called quasinormal modes (QNMs) [1122, 1516–1518]. Here, ωlmn is the characteristicoscillation frequency of the final BH, τlmn is the damping time, Almn is the amplitudeat a distance r, φlmn is the phase, l and m (|m| ≤ l) are angular indices describing howradiation is distributed on the final BH’s sky, and n is an overtone index labeling the(infinite, countable) number of modes.

Equation (38) sets the ground for GW spectroscopy: QNMs are the fingerprints ofthe final BH and extracting their frequency and damping time allows for a consistencycheck of the GW templates [16], tests of gravity [17, 1197, 1506], and of the no-hair properties of the Kerr solution [1519]. As a corollary of the GR uniquenesstheorem [1520, 1521], the entire QNM spectrum of a Kerr BH eventually depends onlyon its mass and spin. Thus, detecting several modes allows for multiple null-hypothesistests of the Kerr metric [1522–1524] and, in turn, of one of the pillars of GR.


3.3.1. Background BH perturbation theory beyond GR is well established in the caseof nonspinning background solutions. Extending seminal work done in GR [1525–1528],the BH metric is approximated as gµν = g(0)

µν +hµν , where g(0)µν is the background solution,

namely the static BH metric that solves the modified Einstein equations in a giventheory, whereas hµν 1 are the metric perturbations. Likewise, all possible extrafields (e.g. the scalar field in a scalar-tensor theory) are linearized around their ownbackground value. Perturbations of spherically symmetric objects can be convenientlydecomposed in a basis of spherical harmonics, and modes with different harmonic indices(l,m) and different parity decouple from each other.

At linear order, the dynamics is described by two systems of ordinary differentialequations: the axial (or odd parity) sector and the polar (or even parity) sector. Whensupplemented by physical boundary conditions at the horizon and at infinity [1122,1517], each resulting linear system defines an eigenvalue problem whose solutions arethe (complex) QNMs, ωlmn−i/τlmn. Remarkably, for a Schwarzschild BH within GR theodd and even parity sectors are isospectral [1528], whereas this property is genericallybroken in other gravity theories.

Perturbations of spinning BHs beyond GR are much more involved. In GR, thegravitational perturbations of a Kerr BH can be separated using the Newman-Penroseformalism [851, 1528] and the corresponding QNMs are described by a single masterequation. This property does not generically hold beyond GR or for other classesof modes. To treat more generic cases, one could perform a perturbative analysisin the spin (the so-called slow-rotation approximation [1529, 1530]) or instead solvethe corresponding set of partial differential equations with spectral methods and otherelliptic solvers [1531]. In general, the spectrum of spinning BHs beyond GR is richerand more difficult to study. This fact has limited the development of parametrizedapproaches to ringdown tests [1532], which clearly require taking into account the spinof the final object.

Remarkably, there is a tight relation between the BH QNMs in the eikonal limit(l 1) and some geodesic properties associated to the spherical photon orbit (thephoton sphere) [1533–1535]. This “null geodesic correspondence” is useful as it requiresonly manipulation of background quantities which are easy to obtain. Furthermore, itprovides a clear physical insight into the BH QNMs in terms of waves trapped within thephoton sphere, slowly leaking out on a timescale given by the geodesic instability timescale. Within this approximation, the QNMs of BHs beyond GR have been recentlystudied in Refs. [1536] and a parametrized approach has been proposed in Ref. [1537].There are however two important limitations. First, the correspondence is strictlyvalid only in the eikonal limit, l 1, whereas the GW signal is typically dominatedby the lowest-l modes. More importantly, the geodesic properties are “kinematical”and do not take into account dynamical aspects, e.g. those related to extra degreesof freedom [1536] or nonminimal couplings [1538]. For example, Schwarzschild BHs intwo different theories of gravity will share the same geodesics but will have in generaldifferent QNM spectra [1536, 1539].


3.3.2. Signatures There are several distinctive features of linearized BH dynamics thatcan be used to perform tests of gravity in the strong-field regime.

Ringdown tests & the no-hair theorem For BBH mergers within GR, the QNMs excitedto larger amplitudes are usually the (2, 2, 0) and (3, 3, 0) gravitational modes [1540–1543]. Because the Kerr BH depends only on two parameters, extracting ω220 and τ220

allows to estimate the mass and spin of the final object, whereas extracting furthersubleading modes provides multiple independent consistency checks of the Kerr metric,since the QNMs are generically different in extensions of GR. Since QNMs probe a highlydynamical aspect of the theory, the latter statement is true even for those theories inwhich the Kerr metric is the only stationary vacuum solution [1539]. These tests requireshigh SNR in the ringdown phase and will be best performed with next generationdetectors, especially with LISA [1125, 1523, 1544, 1545], although coherent mode-stacking methods may be feasible also with advanced ground-based detectors [1546].Furthermore, the ability to perform accurate tests will rely on understanding theoreticalissues such the starting time of the ringdown [1543, 1547] and on the modelling of higherharmonics [1336]. Such tests are based on the prompt ringdown response and implicitlyassume that the remnant object has an horizon. If an ultracompact exotic compactobject rather than a BH forms as a results of beyond-GR corrections, the ringdown signaldepends strongly on the final object’s compactness. If the compactness is comparableor slightly larger than that of a neutron star, the prompt ringdown will show distinctivesignatures [1548–1550]. On the other hand, for objects as compact as BHs, the promptringdown is identical to that of a Kerr BH, but the signal is characterized by “echoes”at late times [1551–1553] (see also Sec. 4.3).

Extra ringdown modes In addition to the shift of gravitational modes discussed above,virtually any extension of GR predicts extra degrees of freedom [1197] which may beexcited during the merger [1108, 1536, 1554, 1555]. Although the amplitude of suchmodes is still poorly estimated, this type of tests calls for novel ringdown searchesincluding two modes of different nature [1556].

Isospectrality Isospectrality of Schwarzschild BHs in GR has a bearing also on KerrBHs. Since this property is generically broken in alternative theories, a clear signatureof beyond-GR physics would be the appearance of a “mode doublet” (i.e., two modeswith very similar frequency and damping time) in the waveform [1532, 1536].

Instabilities BHs in Einstein’s theory are (linearly mode) stable. A notable exceptionrelates to the superradiant instability triggered by minimally-coupled light bosonicfields [1156] (see also Section 5 below). For astrophysical BHs, this instability isineffective except for ultralight bosons with masses below 10−10 eV, in which case ithas important phenomenological consequences (see Sec. 5.7). The situation is differentin modified gravity, for example certain BH solutions are known to be unstable in


scalar-tensor theories in the presence of matter [1557, 1558], in theories with higher-order curvature couplings [1420, 1424], and in massive (bi)-gravity [1559, 1560]. Theseinstabilities are not necessarily pathological, but simply indicate the existence of adifferent, stable, “ground state” in the spectrum of BH solutions to a given theory.Finally, as discussed in Sec. 4.3, extensions of GR might predict exotic compact objectswithout an event horizon. These objects are typically unstable and their instability canlead to peculiar signatures also in the GW signal [1553, 1561].

3.3.3. A parametrised ringdown approach? It should be clear from the previousdiscussion that the Teukolsky formalism is insufficient to study BH perturbation theoryin alternative theories of gravity. Extending it is hence a major open problem. Onemight be even more ambitious and attempt to develop a theory-agnostic parametrisationfor the ringdown, similar in spirit to that discussed in Sec. 3.2 for the inspiral phase.This would generally be a two-faceted problem, since the quiescent BH that acts asthe endpoint of evolution is not a Kerr BH, in general. Hence, one first needs toparametrise deviations from the Kerr spacetime and then propose a meaningful andaccurate parametrisation of the perturbations around such non-Kerr background.

There have been several attempts to parametrise stationary deviations fromthe Kerr spacetime. Bumpy metrics were constructed in Refs. [1562–1564] withinGR, starting from a perturbation around the Schwarzschild metric and applyingthe Newman-Janis transformation [1565] to construct a rotating configuration. Thequasi-Kerr metric uses the Hartle-Thorne metric [1566], valid for slowly-rotatingconfigurations, and modifies the quadrupole moment [1567]. Non-Kerr metrics canalso be found, with the extra requirement that a Carter-like, second-order Killing tensorexists [1568]. This approach and metric were later simplified in Ref. [1569]. Note howeverthat rotating BH solutions in alternative theories of gravity do not possess such a Killingtensor in general. Johannsen and Psaltis [1570] proposed the modified Kerr metric bystarting from a spherically symmetric and static metric (that does not necessarily satisfythe Einstein equations) and applying the Newman-Janis transformation. This approachwas later extended in Ref. [1571]. Finally, modified axisymmetric spacetime geometriescan be found by adopting a continued-fraction expansion to obtain a rapid convergencein the series [1572, 1573]. See e.g. Refs. [1574, 1575] for other parameterizations fordeviations from Kerr. Properties of various parameterized Kerr metrics have beenstudied in [1576]. See also Ref. [1577] for more details on such generic parameterizationof the Kerr metric.

It should be stressed that a shortfall of all of the aforementioned parametrisationis that they do not have a clear physical interpretation and this makes them rather adhoc. Moreover, they only address half of the problem of parametrizing the ringdown.So far their use in modelling waveforms has been limited, but see Refs. [1373, 1578]and Ref. [1537] for applications to extreme-mass ratio inspirals and BH ringdownrespectively. At this stage there has been very little progress on how one can take thesecond step, i.e. parametrise linear perturbations around such spacetimes in a theory-


agnostic way. Along these lines, a general theory of gravitational perturbations of aSchwarzschild metric has been developed in Ref. [1554] and applied to Horndeski theoryin Ref. [1555].

3.4. Merger, numerics, and complete waveforms beyond General RelativityContributors: C. Palenzuela, T. P. Sotiriou

The merger of binary compact objects will test the highly dynamical andstrongly non-linear regime of gravity, that can only be modelled by using numericalsimulations. As already discussed above, the recent direct detection of GWs from BBHs[1, 2, 11, 18, 20], and of the NSB merger [21] with an associated short GRB [23] anda plethora of concurrent electromagnetic signals from the same source [22] has alreadyprovided constraints on deviations from GR in various forms and contexts (e.g. [17, 674–677, 1491, 1492, 1506] respectively. However, most of these constraints use either partial,potentially parametrised, waveforms or rely on propagation effects. In fact, the truepotential of testing GR is currently limited by the lack of knowledge of GW emissionduring the merger phase in alternatives to GR [1506]. This problem is particularly acutefor heavier BH mergers, where only a short part of the inspiral is detected.

This suggests that constraints could be strengthened significantly in most cases ifone had complete, theory-specific, waveforms. However, performing stable and accuratenumerical simulations that would produce such waveforms requires an understandingof several complex issues. Probably the first among them is the well-posedness of thesystem of equations that describe evolution of a given alternative scenario to GR. In thenext section we briefly describe the issue of well-posedness and discuss some techniquesthat have been used so far to obtain a well-posed evolution system in alternative theories.In Sec. 3.4.3 we overview some recent numerical studies of nonlinear evolution beyondGR.

3.4.1. Initial value formulation and predictivity beyond GR Modelling the evolution ofa binary system for given initial data is a type of initial value problem (IVP). An IVPis well-posed if the solution exists, is unique and does not change abruptly for smallchanges in the data. A theory with an ill-posed IVP cannot make predictions. The IVPis well-posed in GR [1579] but it is not clear if this is true for most of its contenders. Thisis a vastly overlooked issue and a systematic exploration of the IVP in many interestingalternative theories, such as those discussed in Sec. 2.1, is pending.

A class of alternative theories of gravity that are known to be well-posed is scalar-tensor theories described by action (21). As discussed in Sec. 2.2, after suitable fieldredefinitions, these theories take the form of GR plus a scalar field with a canonicalkinetic term and potential non-minimal couplings between the scalar and standard modelfields, see action (22). Since these couplings do not contain more than two derivatives,one can argue for well-posedness using the known results for GR and the fact that lower-order derivative terms are not relevant for this discussion. Interestingly, most alternative


theories actually include modifications to the leading order derivative terms in themodified Einstein’s equations, so a theory-specific study is necessary. This has beenattempted in very few cases and results are mostly very preliminary. In particular, thereis evidence that dynamical Chern-Simons gravity is ill-posed [1110]. Certain generalisedscalar-tensor theories appear to be ill-posed in a generalised harmonic gauge whenlinearised over a generic, weak field background [1580]; however, note that this result isgauge-dependent, and hence not conclusive. Finally, in certain Lorentz-violating theoriesthat exhibit instantaneous propagation, describing evolution might require solving amixture of hyperbolic and elliptic equations (where the latter are not constraints as inGR) [1486, 1487].

3.4.2. Well-posedness and effective field theories One is tempted to use well-posednessas a selection criterion for alternative theories of gravity, as a physical theory certainlyneeds to be predictive (in an appropriate sense). However, alternatives to GR can bethought of as effective field theories — truncations of a larger theory and hence inherentlylimited in their range of validity. This complicates the question of what one should dowhen a given theory turns out to be ill-posed. In fact, it even affects how one shouldview its field equations and the dynamics they describe in the first place. Effective fieldtheories (EFTs) can often contain spurious degrees of freedom (e.g. ghosts) that leadto pathological dynamics. In linearised theories it is easy to remove such degrees offreedom and the corresponding pathologies, but there is no unique prescription to doingso in general. Hence, instead of setting aside theories that appear to be ill-posed whentaken at face value, perhaps one should be looking for a way to ‘cure’ them and renderthem predictive at nonlinear level.

A very well-known EFT, viscous relativistic hydrodynamics, requires such ‘curing’to control undesirable effects of higher order derivatives and a prescription for doingso has been given long ago [1581–1583]. A similar prescription applicable to gravitytheories has been given recently [1584]. Roughly speaking, this approach treats thetheory as a gradient expansion and, hence, it considers as the cause of the pathologiesrunaway energy transfer to the ultraviolet, that in turn renders the gradient expansioninapplicable. As a results, it attempts to modify the equations so as to prevent suchtransfer and to ensure that the system remains within the regime of validity of theeffective descriptions throughout the evolution.

Another approach is to consider the theory as arising from a perturbative expansionis a certain parameter, say λ. If that were the case, then higher order corrections in λhave been neglected and, consequently, the solutions can only be trusted only up to acertain order in λ. Moreover, they have to be perturbatively close (in λ) to solutions withλ = 0, which are solution of GR. Hence, one can iteratively generate fully dynamicalsolutions order-by-order in λ. This process is effectively an order-reduction algorithmand yields a well-posed system of equations. It has a long history in gravity theories indifferent contexts (e.g. [1585–1587]), but it has only recently been used for nonlinear,dynamical evolution in alternative theories [1108, 1588, 1589].


These two ‘cures’ do not necessarily give the same results. Moreover, there is noreason to believe that there cannot be other ways to address this problem, so this remainsa crucial open question. In principle, the way that an EFT is obtained from a morefundamental theory should strongly suggest which is the way forward when consideringnonlinear evolution. However, in practice one often starts with an EFT and hopes toeventually relate it to some fundamental theory. Hence, it seems wise to consider allpossible approaches. In principle different theories might require different approaches.

3.4.3. Numerical simulations in alternative theories As mentioned above, one canstraightforwardly argue that the IVP is well-posed in standard scalar-tensor theoriesdescribed by action (21). However, as will be discussed in detail in Sec. 4.1, BHs inthis class of theories are generically identical to GR and carry no scalar configuration.Hence, even though gravitational radiation in these theories can in principle contain alongitudinal component, it is highly unlikely it will get excited in a BBH. In Sec. 4.2we outline the conditions under which BHs can differ from their GR counterparts instandard scalar-tensor theories.∗ When these conditions are satisfied there should bean imprint in GWs from BH binaries. It is worth highlighting here a couple of caseswhere numerical simulations have to be used to address this question. The first has todo with the role of asymptotics. It has been shown that time-dependent asymptoticfor the scalar could lead to scalar radiation during the coalescence [1107], though thisemission would probably be undetectable for realistic asymptotic values of the scalarfield gradient. The second case has to do with whether matter in the vicinity of a BH canforce it to develop a non-trivial configuration. This has been shown in idealised setupsonly in Refs. [1557, 1558]. However, numerical simulations for the same phenomenonin NSs have been performed in [1226, 1227, 1399].

f(R) theories of gravity (see Ref. [1590] for a review), which are dynamicallyequivalent to a specific subclass of scalar-tensor theories, are also well-posed [1590, 1591].A comparative study of NSB mergers in GR with those of a one-parameter model off(R) = R + aR2 gravity is performed in [1592].

A well-posed extension of scalar-tensor theories is Einstein-Maxwell-Dilaton gravity.It has its origin in low energy approximations of string theory and it includes, apartfrom the metric and a scalar, a U(1) gauge field. The scalar field couples exponentiallyto the gauge field, allowing for deviations from GR even for BHs with asymptoticallyconstant scalar field. BBH simulations have shown that these deviations are rathersmall for reasonable values of the hidden charge [1515], leading to weak constraints onthe free parameters of the theory.

Finally, some first numerical results have recently appeared in scalar-Gauss–Bonnetgravity, and specifically the theory described by action (30), and in Chern-Simons gravity(see Sec. 2.2 for more details on the theories). Refs. [1588, 1589] studied scalar evolutionin scalar-Gauss–Bonnet gravity and Ref. [1108] performed the first binary evolution in

∗ It might hence be preferable for readers that are not familiar with the literature on hairy BHs to returnto this section after going through Secs. 4.1 and 4.2.


Chern-Simons gravity. These results are notable for using a perturbative expansion inthe free parameter of the theory, described in the previous section, in order to circumventpotential issues with well-posedness (as discussed above, if the field equation are takenat face value, well-posedness is known to be an issue for Chern-Simons gravity [1110]and it might be an issue for scalar-Gauss–Bonnet gravity [1580]).

4. The nature of Compact Objects

4.1. No-hair theoremsContributors: C. Herdeiro, T. P. Sotiriou

The discovery of the Schwarzschild solution in 1915-16 [1593], shortly afterEinstein’s presentation of GR [1594], triggered a half-a-century long quest for its rotatingcounterpart, which ended with the discovery of the Kerr solution in 1963 [1595] -see [1596] for a historical account of this discovery. At this time, unlike in Schwarzschild’sepoch, it was already considered plausible that these metrics could represent theendpoint of the gravitational collapse of stars, even though the name black hole todescribe them only became widespread after being used by Wheeler in 1967-68 [1597].

The extraordinary significance of the Kerr metric became clear with theestablishment of the uniqueness theorems for BHs in GR - see [1380] for a review. Thefirst such theorem was that of Israel’s for the static case [1598]. Then, Carter [1520] andRobinson [1599] constructed a theorem stating: An asymptotically flat stationary andaxisymmetric vacuum spacetime that is non-singular on and outside a connected eventhorizon is a member of the two-parameter Kerr family. The axial symmetry assumptionwas subsequently shown to be unnecessary (assuming analyticity, nondegenerary, andcausality [1600]); for BHs, stationarity actually implies axial symmetry, via the “rigiditytheorem,” which relates the teleologically defined “event horizon” to the locally defined“Killing horizon” [1601]. The upshot of these theorems is that a stationary vacuum BH,despite possessing an infinite number of non-trivial, appropriately defined multipolarmoments [1602], has only two degrees of freedom: its mass and angular momentum.This simplicity was written in stone by Wheeler’s dictum “BHs have no hair” [1603].

In parallel to these developments the astrophysical evidence for BHs piled up overthe last half century - see e.g. [1604]. In fact, the very conference where the Kerrsolution was first presented (the first Texas symposium on relativistic astrophysics) heldin December 1963, was largely motivated by the discovery of quasars in the 1950s and thegrowing belief that relativistic phenomena related to highly compact objects should beat the source of the extraordinary energies emitted by such very distant objects [1605].Gravitational collapse was envisaged to play a key role in the formation of thesehighly compact objects and such collapse starts from matter distributions, rather thanvacuum. Thus, one may ask if BHs in the presence of matter, rather than in vacuum,still have no-hair. In this respect, Wheeler’s dictum was a conjecture, rather than asynthetic expression of the uniqueness theorems. It hypothesised a much more general


and ambitious conclusion than that allowed solely by these theorems. The conjecturestated that: gravitational collapse leads to equilibrium BHs uniquely determined by mass,angular momentum and electric charge -asymptotically measured quantities subject to aGauss law and no other independent characteristics (hair) [1606]. Here, the Gauss lawplays a key role to single out the physical properties that survive gravitational collapse,because they are anchored to a fundamental (gauged) symmetry, whose conserved chargecannot be cloaked by the event horizon. The uniqueness theorems (which can begeneralised to electro-vacuum [1380]), together with the no-hair conjecture, lay down thefoundation for the Kerr hypothesis, that the myriad of BHs that exist in the Cosmos arewell described, when near equilibrium, by the Kerr metric. This astonishing realisationwas elegantly summarised by S. Chandrasekhar [1607]: In my entire scientific life,extending over forty-five years, the most shattering experience has been the realisationthat an exact solution of Einstein’s field equations of GR, discovered by the New Zealandmathematician, Roy Kerr, provides the absolutely exact representation of untold numbersof massive BHs that populate the Universe.

The Kerr hypothesis can be tested both theoretically and by confrontation withobservations. On the theory front, the key question is whether BHs that are notdescribed by the Kerr metric can constitute solutions of reasonable field equations thatdescribe gravity and potentially other fields. Additionally, one can ask whether suchBH solutions can form dynamically and are (sufficiently) stable as to be astrophysicallyrelevant. On the observational front, the question is to which extent the BHs we observeare compatible with the Kerr hypothesis and what constraints can one extract from that.

A set of results on the non-existence of non-Kerr BHs in specific models, eitherincluding different matter terms in Einstein’s GR or by considering theories of gravitybeyond Einstein’s GR (or both) have been established since the 1970s and are knownas no-hair theorems - see e.g. the reviews [1407, 1608–1610].∗ No-hair theorems applyto specific models or classes of models and rely on assumptions regarding symmetries,asymptotics, etc. As will become evident in the next section, dropping one or moreof these assumptions in order to circumvent the corresponding no-hair theorem canbe a good tactic for finding non-Kerr BHs (see also Ref. [1407] for a discussion). Moregenerally, understanding the various sort of theorems, their assumptions and limitationsis instructive.

A large body of work was dedicated to the case of scalar hair, partly due tothe conceptual and technical simplicity of scalar fields. Indeed, one of the earliestexamples of a no-hair theorem was provided by Chase [1611], following [1612]. Chaseinquired if a static, regular BH spacetime could be found in GR minimally coupledto a real, massless scalar field. Without assuming spherical symmetry, he could showthat the scalar field necessarily becomes singular at a simply-connected event horizon.

∗ Often, the uniqueness theorems are also called no-hair theorems, under the rationale that they showthat in vacuum (or electro-vacuum), BHs have no independent multipolar hair, and only two (or three,ignoring magnetic charge) degrees of freedom. Here we use no-hair theorems to refer to no-go resultsfor non-Kerr BHs in models beyond electro-vacuum GR.


Thus, a static BH cannot support scalar hair, in this model. The 1970s witnessed theformulation of influential no-hair theorems, notably, by Bekenstein, who developed adifferent approach for establishing the non-existence of scalar hair, applicable to higherspin fields as well [1613–1615]. An elegant no-hair theorem was given by Hawking [1404]for minimally coupled scalar fields without self interactions and for Brans-Dicke theory[1199]. Hawking’s theorem assumes stationarity (as opposed to staticity) and no othersymmetry and its proof is particularly succinct. This theorem has been later generalisedby Sotiriou and Faraoni [1406] to scalars that exhibit self interactions and to the classof scalar-tensor theories described by action (21). Hui and Nicolis [1411] have provena no-hair theorem for shift-symmetric (and hence massless) scalar fields that belong tothe Horndeski class discussed in Sec. 2.2. This proof assumes staticity and sphericalsymmetry, though it can be easily generalized to slowly-rotating BHs [1412]. It is alsosubject to a notable exception, as pointed out in Ref. [1412]: a linear coupling betweenthe scalar and the Gauss-Bonnet invariant is shift symmetric and yet circumvents thetheorem [see Sec. 2.2 and action (30)]. Finally, a no-hair theorem has been given inRef. [1420] for stationary BH solutions for the theory described by action (31), providedf ′′(φ0)G < 0; a similar proof, but restricted to spherical symmetry, can be found inRef. [1421].

4.2. Non-Kerr black holesContributors: C. Herdeiro, T. P. Sotiriou

Despite the many no-hair theorems discussed above, hairy BH solutions are knownto exist in many different contexts. In GR, the 1980s and 1990s saw the construction ofa variety of hairy BH solutions, typically in theories with non-linear matter sources - seee.g. the reviews [1608, 1616, 1617]. The paradigmatic example are the coloured BHs,found in Einstein-Yang-Mills theory [1618–1621]. However, these BHs are unstable andhave no known formation mechanism. Hence, they should not be considered as counterexamples to the weak no-hair conjecture, and it is unlikely that they play a role in theastrophysical scene.

Below we will review some more recent attempt to find hairy BH solutions thatare potentially astrophysically relevant. We first focus on cases that are covered byno-hair theorems and discuss how dropping certain assumptions can lead to hairy BHs.We then move on to various cases of hairy BHs in theories and models that are simplynot covered by no-hair theorems. This is not meant to be an exhaustive review butmerely a discussion of some illuminating examples in order to highlight the interestingBH physics that GWs could potentially probe.

4.2.1. Circumventing no-hair theorems No-hair theorems rely on assumptions, suchas: (i) the asymptotic flatness of the metric and other fields; (ii) stationarity (or morerestrictive symmetries); (iii) the absence of matter; (iv) stability, energy conditions andother theorem-specific conditions. There is no doubt that removing any one of these


assumptions leads to BH hair. For example, standard scalar-tensor theories describedby the action (21) are covered by the no-hair theorems discussed in the previous section[1404, 1406, 1611, 1613–1615]. Nonetheless, BHs with scalar hair exist in these theories ifone allows for anti-de Sitter (AdS) asymptotics (e.g. Ref. [1622]), or attempts to “embed”the BHs in an evolving universe [1623], or allows for matter to be in their vicinity[1557]. The case of AdS asymptotics, though interesting theoretically, is not relevantfor astrophysics. The fact that cosmic evolution could potentially endow astrophysicalBHs with scalar hair is certainly enticing but the effect should be very small [1624].Finally, it is not yet clear if the presence of matter in the vicinity of a BH could inducea scalar charge that is detectable with the current observations. See Ref. [1407] for amore detailed discussion. Circumventing the theorems by violating stability argumentswill be discussed in Sec. 4.2.3.

A perhaps less obvious way to obtain hairy solutions is to strictly impose thesymmetry assumptions on the metric only and relax them for other fields. Even thoughthe field equations relate the fields their symmetries do not need to match exactlyin order to be compatible (see, for instance, Ref. [1625] for a recent discussion). Awell-studied case is that of a complex, massive scalar field minimally coupled to gravity[1168, 1626]. The scalar has a time-dependent phase, but its stress-energy tensor remainstime-independent (thanks to some tuning) and the metric can remain stationary. Seealso [1627, 1628] for generalisations and [1172] for an existence proof of the solutions.BHs with Proca (massive vector) hair have also been found using the same approach[1171].

Whether or not these hairy BH are relevant for astrophysical phenomena depends onthree main factors: (1) the existence of (ultra-light) massive bosonic fields in Nature; (2)the existence of a formation mechanism and (3) their stability properties. The first factoris an open issue. Ultra-light bosonic fields of the sort necessary for the existence of theseBHs with astrophysical masses do not occur in the standard model of particle physics.Some beyond the standard model scenarios, however, motivate the existence of this typeof “matter,” notably the axiverse scenario in string theory [1629]. The second point hasbeen positively answered by fully non-linear numerical simulations [1166, 1630]. Thereis at least one formation channel for these BHs, via the superradiant instability of KerrBHs, in the presence of ultralight bosonic fields - see [1156] for a review of this instability.The hairy BHs that result from this dynamical process, however, are never very hairy,being always rather Kerr-like. Concerning the third point, it has been recognisedsince their discovery, that these BHs could be afflicted by the superradiant instabilitythemselves [1626, 1631]. Recent work [1632] succeeded in computing the correspondingtimescales, and reported that the timescale of the strongest superradiant instability thatcan afflict hairy BH is roughly 1000 times longer than that of the strongest instabilitythat can afflict the Kerr BH. This study, albeit based on the analysis of a very smallsample of solutions, leaves room for the possibility that some of these BHs can formdynamically and be sufficiently stable to play a role in astrophysical processes [1173].However, a more detailed analysis needs to be performed before definite conclusions can


be drawn.Hairy BHs have also been found in certain shift-symmetric scalar tensor theories

by allowing the scalar to depend linearly on time while requiring that the metric bestatic [1633, 1634]. This is possible because shift symmetry implies that the scalaronly appears in the equations through its gradient and the linear time dependencerenders the gradient time independent. The linear stability on such solutions has beenexplored in Refs. [1635–1637]. It remains largely unexplored whether these BHs canform dynamically and hence whether they are relevant for astrophysics.

4.2.2. Black holes with scalar hair The most straightforward way to find hairyBH solutions is to consider theories that are not covered by no-hair theorems. Inthe case of scalar-tensor theories, it is known that couplings between the scalar (orpseudoescalar) and quadratic curvature invariants, such as the Gauss-Bonnet invariantG ≡ R2 − 4RµνR

µν + RµνρσRµνρσ or the Pontryagin density ∗Rα γδ

β Rβαγδ, can lead to

scalar hair, see e.g. [1412, 1414–1417]. Indeed, the existence of hairy BHs singles outspecific class of theories, such as dynamical Chern-Simons (dCS) gravity [1437, 1438] andthe scalar-Gauss-Bonnet theories described by the action (31) of Sec. 2.2 (which havehairy BH solutions provided that f ′(φ0) 6= 0 for any constant φ0 [1412, 1420]). Sec. 2.2already contains a discussion about these theories and their BH phenomenology, so werefer the reader there for details. See also Sec. 3.4.3 for a discussion on the first attemptsto simulate dynamical evolution of hairy BHs and to produce waveforms for binariesthat contain them.

4.2.3. Black hole scalarization In the previous section the focus was on theories thatare not covered by no hair theorems and have hairy BH for all masses. An interestingalternative was recently pointed out in Refs. [1420, 1422]: that certain theories thatmarginally manage to escape no-hair theorems [1420, 1421] can have hairy BHs only incertain mass ranges, outside which BHs are identical to those of GR. The theories belongin the class describe by action (31) and the phenomenon resembles the“spontaneousscalarization” of stars that happens in certain models of the standard scalar-tensortheories of action (21); see Sec. 2.2 for more details. In the mass ranges where hairyBHs exist the GR BHs of the same mass are expected to be unstable and this instabilityis supposed to gives rise to the scalar hair. However, depending on the model, the hairysolutions can also be unstable [1424] and this issue requires further investigation. It isalso not clear if other theories can also exhibit BH scalarization and the astrophysicalsignificance of the effect has not yet be explored.

4.2.4. Black holes in theories with vector fields and massive/bimetric theories Inanalogy to the generalised scalar-tensor theories discussed in Sec. 2.2 and described byaction (25), one can construct the most general vector-tensor theories with second-orderequations of motion. These theories can be considered as generalised Proca theories andthey contain new vector interactions [1638, 1639]. Their phenomenology can be distinct


from that of Horndeski scalars and BH solutions with vector hair are known to exist[1640–1646]. Depending on the order of derivative interactions, these BH solutions canhave primary (i.e. the new charge is independent) or secondary (i.e. the new charge isnot independent) Proca hair. The presence of a temporal vector component significantlyenlarges the possibility for the existence of hairy BH solutions without the need of tuningthe models [1642, 1643].

Horndeski scalar-tensor and the generalised Proca theories can be unified intoscalar-vector-tensor theories for both the gauge invariant and the gauge broken cases[1647]. In the U(1) gauge invariant and shift symmetric theories the presence of cubicscalar-vector-tensor interactions is crucial for obtaining scalar hair, which manifestsitself around the event horizon. The inclusion of the quartic order scalar-vector-tensorinteractions enables the presence of regular BH solutions endowed with scalar and vectorhairs [1648]. It is worth mentioning that this new type of hairy BH solutions are stableagainst odd-parity perturbations [1649].

In massive and bimetric gravity theories the properties of the BH solutions dependheavily on choices one makes for the fiducial or dynamical second metric [1650–1655].If the two metrics are forced to be diagonal in the same coordinate system (bidiagonalsolutions) their horizons need to coincide if they are not singular [1656]. It should bestressed that generically there is not enough coordinate freedom to enforce the metricsto be bidiagonal. The analysis of linear perturbations reveals also an unstable modewith a Gregory-Laflamme instability [1559, 1560], which remains in bi-Kerr geometry.Perturbations of the non-bidiagonal geometry is better behaved [1657]. Non-singularand time dependent numerical solutions were recently found in [1658].

4.2.5. Black holes in Lorentz-violating theories Lorentz symmetry is central to thedefinition of a BH thanks to the assumption that the speed of light is the maximumattainable speed. Superluminal propagation is typical of Lorentz violating theories∗ andGW observations have already given a spectacular constraint: the detection of a NSBmerger (GW170817) with coincident gamma ray emission has constrained the speedof GWs to a part in 1015 [23]. However, Lorentz-violating theories generically exhibitextra polarizations [1484], whose speed remains virtually unconstrained [1491]. See alsoSec. 3.1. Moreover, BHs in such theories are expected to always be hairy. The reasonhas essentially been explained in the discussion of Sec. 2.3: Lorentz-violating theoriescan generally be written in a covariant way (potentially after restoring diffeomorphisminvariance through the introduction of a Stueckelberg field) and in this setup Lorentzsymmetry breaking can be attributed to some extra field that has the property of beingnontrivial in any solution to the field equation. That same property will then endowBHs with hair. Indeed, this is known to be true for the theories reviewed in Sec. 2.3.The structure of such BHs depends on the causal structure of the corresponding theory.

In particular, in Einstein-aether theory (æ-theory), though speeds of linear

∗ It might be worth stressing that superluminal propagation does not necessarily lead to causalconundrums in Lorentz-violating theories.


perturbations can exceed the speed of light, they satisfy an upper bound set by the choiceof the ci parameters. Indeed, massless excitations travel along null cones of effectivemetrics composed by the metric that defines light propagation and the aether [1659].This makes the causal structure quasi-relativistic, when one adopts the perspective ofthe effective metric with the widest null cone [1487]. Stationary BHs turn out to havemultiple nested horizons, each of which is a Killing horizon of one of the effective metricsand acts as a causal boundary of a specific excitation [1659, 1660].

In Hořava gravity there is no maximum speed, as already explained is Sec. 2.3 .There is a preferred foliation and all propagating modes have dispersion relations thatscale as ω2 ∝ k6 for large wave numbers k and there is arbitrarily fast propagation.Even at low momenta, where the dispersion relations are truncated to be linear, there isalso an instantaneous (elliptic) mode, both at perturbative [1485] and nonperturbativelevel [1486]. It would be tempting to conclude that BHs cannot exist in Hořava gravity.Nonetheless, there is a new type of causal horizon that allows one to properly define aBH, dubbed the universal horizon [1485, 1660]. It is not a null surface of any metric,but a spacelike leaf of the preferred foliation that cloaks the singularity. Future-directedsignals can only cross it in one direction and this shields the exterior from receivingany signal, even an instantaneous one, from the interior. Universal horizons persist inrotating BHs in Hořava gravity [1487, 1488, 1661–1663] and have been rigorously definedwithout resorting to symmetries in Ref. [1487].

Even though BHs in Lorentz-violating theories and their GR counterparts have verydifferent causal structure, their exteriors can be very similar and hard to tell apart withelectromagnetic observations [1488, 1660, 1663]. Nevertheless, the fact that Lorentz-violating theories have additional polarisations and their BHs are have hair suggestsvery strongly that binary systems will emit differently than in GR. Hence, confrontingmodel-dependent waveforms with observations is likely to yields strong constraints forLorentz violation in gravity.

4.3. Horizonless exotic compact objectsContributors: V. Cardoso, V. Ferrari, P. Pani, F. H. Vincent

BHs are the most dramatic prediction of GR. They are defined by an event horizon,a null-like surface that causally disconnects the interior from our outside world. Whileinitially considered only as curious mathematical solutions to GR, BHs have by nowacquired a central role in astrophysics: they are commonly accepted to be the outcomeof gravitational collapse, to power active galactic nuclei, and to grow hierarchicallythrough mergers during galaxy evolution. The BH paradigm explains a variety ofastrophysical phenomena and is so far the simplest explanation for all observations.It is, however, useful to remember that the only evidence for BHs in our universe is theexistence of dark, compact and massive objects. In addition, BHs come hand in handwith singularities and unresolved quantum effects. Hence, it is useful to propose andto study exotic compact objects (ECOs) – dark objects more compact than NSs but


without an event horizon.Perhaps the strongest theoretical motivation to study ECO’s follows from quantum

gravity. Quantum modifications to GR are expected to shed light on theoretical issuessuch as the pathological inner structure of Kerr BHs and information loss in BHevaporation. While there is a host of recent research on the quantum structure ofBHs, it is not clear whether these quantum effects are of no consequence on physicsoutside the horizon or will rather lead to new physics that resolves singularities anddoes away with horizons altogether. The consequence of any such detection, howeversmall the chance, would no less than shake the foundations of physics. As there is nocomplete quantum gravity approach available yet, the study of ECOs, in various degreesof phenomenology, is absolutely crucial to connect to current GW observations.

In summary, ECOs should be used both as a testing ground to quantify theobservational evidence for BHs, but also to understand the signatures of alternativeproposals and to search for them in GW and electromagnetic data [1553].

Classification: Compact objects can be conveniently classified according to theircompactness, M/R, where M and R are the mass and (effective) radius of the object,respectively. NSs have M/R ≈ 0.1− 0.2, a Schwarzschild BH has M/R = 1/2, whereasa nearly extremal Kerr BH has M/R ≈ 1. The study of the dynamics of light raysand of GWs shows that the ability of ECOs to mimic BHs is tied to the existence ofan unstable light ring in the geometry [1553, 1664, 1665]. Accordingly, two importantcategories of this classification are [1553, 1665]:

(i) Ultracompact objects (UCOs), whose exterior spacetime has a photon sphere. Inthe static case, this requires M/R > 1/3. UCOs are expected to be very similarto BHs in terms of geodesic motion. NSs are not in this category, unless they areunrealistically compact and rapidly spinning.

(ii) Clean-photon-sphere objects (ClePhOs), so compact that the light travel time fromthe photon sphere to the surface and back is longer than the characteristic timescale, τgeo ∼ M , associated with null geodesics along the photon sphere. Theseobjects are therefore expected to be very similar to BHs at least over dynamicaltime scales ∼M . In the static case, this requires M/R > 0.492 [1553, 1665].

Some models of ECOs have been proposed in attempts to challenge the BHparadigm. Others are motivated by (semiclassical) quantum gravity scenarios thatsuggest that BHs would either not form at all [1665–1672], or that new physics shoulddrastically modify the structure of the horizon [1673–1675].

In the presence of exotic matter fields (e.g. ultralight scalars), even classicalGR can lead to ECOs. The most characteristic case is perhaps that of bosonstars [1175, 1676, 1677]: self-gravitating solutions of the Einstein-Klein-Gordon theory.They have no event horizon, no hard surface, and are regular in their interior. Dependingon the scalar self-interactions, they can be as massive as supermassive compact objectsand – when rapidly spinning – as compact as UCOs. Further details about these objectscan be found elsewhere [1678]. Though boson stars are solutions of a well defined theory,


most ECO-candidates currently arise from phenomenological considerations and employsome level of speculation. For instance, certain attempts of including quantum gravityeffects in the physics of gravitational collapse postulate corrections in the geometry ata Planck distance away from the horizon, regardless of the mass (and, hence, of thecurvature scale) of the object. Since lPlanck M , these objects naturally classify asClePhOs. One example are fuzzballs, ensembles of horizonless microstates that emergeas rather generic solutions to several string theories [1667, 1668]. In these solutions theclassical horizon arises as a coarse-grained description of the (horizonless) microstategeometries. Another example are so-called gravitational-vacuum stars, or gravastars,ultracompact objects supported by a negative-pressure fluid [1666, 1679–1681], whichmight arise from one-loop effects in curved spacetime in the hydrodynamical limit [1682].Other examples of ClePhOs inspired by quantum corrections include black stars [1683],superspinars [1684], collapsed polymers [1685, 1686], 2− 2-holes [1687], etc [1665]. Weshould highlight that for most of these models, 1 − 2M/R ∼ 10−39 − 10−46 for stellarto supermassive dark objects. Thus, their phenomenology is very different from that ofboson stars, for which 1− 2M/R ∼ O(0.1) at most.

The speculative nature of most ECOs implies that their formation process hasnot been consistently explored (with notable exceptions [1678]). If they exist theyare likely subject to the so-called ergoregion instability. This was first found inRef. [1688] for scalar and electromagnetic perturbations (see also [1689, 1690]), thenshown to affect gravitational perturbations as well in Ref. [1691], and recently provedrigorously in Ref. [1692]). The instability affects any spinning compact object with anergoregion but without a horizon and its time scale may be shorter than the age ofthe Universe [1693, 1694]. The endpoint of the instability is unknown but a possibilityis that the instability removes angular momentum [1695], leading to a slowly-spinningECO [1156], or perhaps it is quenched by dissipation within the object [1696] (althoughthe effects of viscosity in ECOs are practically unknown). An interesting question isalso the outcome of a potential ECO coalescence. All known equilibrium configurationsof ECOs have a maximum mass above which the object is unstable against radialperturbations and is expected to collapse. The configuration with maximum mass isalso the most compact (stable) one. Therefore, the more an ECO mimics a BH, thecloser it is to its own maximum mass. Thus, if two ECOs are compact enough as toreproduce the inspiral of a BH coalescence (small enough deviations in the multipolemoments, tidal heating, and tidal Love numbers relative to the BH case; see below), theirmerger would likely produce an ECO with mass above the critical mass of the model.Hence the final state could be expected to be a BH. In other words, one might face a“short blanket” problem: ECOs can mimic well either the post-merger ringdown phaseof a BH or the pre-merger inspiral of two BHs, but they may find it difficult to mimicthe entire inspiral-merger-ringdown signal of a BH coalescence with an ECO+ECO →ECO process. The only way out of this problem is to have a mass-radius diagram thatresembles that of BHs for a wide range of masses. No classical model is known thatbehaves this way, yet.


It is worth noting that, even if the BH paradigm is correct, ECOs might be lurkingin the universe along with BHs and NSs. Modelling the EM and GW signatures ofthese exotic sources is necessary to detect them, and may even provide clues for otheroutstanding puzzles in physics, such as DM (see also Section 5 below).

4.4. Testing the nature of compact objectsContributors: V. Cardoso, V. Ferrari, M. Kramer, P. Pani, F. H. Vincent, N. Wex

4.4.1. EM diagnostics EM radiation emitted from the vicinity of BH candidates canprobe the properties of spacetime in the strong-field region and may lead to constrainingthe nature of the compact object. We focus our discussion on three popular EM probes,namely shadows, X-ray spectra, and quasi-periodic oscillations. We do not discuss herepolarization, nor effects on stellar trajectories. Because all EM tests can be eventuallytraced back to geodesic motion, EM probes may distinguish between BHs and ECOswith M/R < 0.492, whereas it is much more challenging to tell a ClePhO from a BHthrough EM measurements.

Shadows BHs appear on a bright background as a dark patch on the sky, due tophotons captured by the horizon. This feature is known as the BH shadow [1697, 1698].The Event Horizon Telescope [1699], aiming at obtaining sub-millimeter images of theshadow of the supermassive object Sgr A* at the center of the Milky Way and of thesupermassive object at the center of the elliptical galaxy M87, is a strong motivationfor these studies.

A rigorous and more technical definition of the BH shadow is the following [1700]:it is the set of directions on an observer’s local sky that asymptotically approach theevent horizon when photons are ray traced backwards in time along them. Thus, by thisdefinition, shadows are intrinsically linked with the existence of a horizon and, strictlyspeaking, an ECO cannot have a shadow. However, in practice ultracompact horizonlessobjects (in particular UCOs and ClePhOs) might be very efficient in mimicking theexterior spacetime of a Kerr BH. It is therefore interesting to study photon trajectoriesin such spacetimes and to contrast them with those occurring in the Kerr case.

In the analysis of shadows, one generally either considers parametrizedspacetimes [1570, 1701–1703] (that allow to tune the departure from Kerr but mightnot map to known solutions of some theory), or takes into account a specific alternativetheoretical framework [1704–1711] or a particular compact object within GR [1712–1715]. Most of these studies obtain differences with respect to the standard Kerrspacetime that are smaller than the current instrumental resolution (< 10µas). Somestudies report more noticeable differences in the case of naked singularities [1712], exoticmatter that violates some energy condition [1713], or in models that allow for large valuesof the non-Kerrness parameters [1708, 1709]. Moreover, demonstrating a clear differencein the shadow is not enough to infer that a test can be made, since such difference might


be degenerate with the mass, spin, distance, and inclination of the source. Finally,the fact that horizonless objects lead to shadow-like regions that can share some clearresemblance with Kerr [1715] shows the extreme difficulty of an unambiguous testbased on such observables (for perspective tests with current observations see, e.g.,Refs. [1716, 1717]).

Kα iron line and continuum-fitting The X-ray spectra of BHs in X-ray binaries andAGNs are routinely used to constrain the spin parameter of BHs, assuming a Kerrmetric [1718, 1719]. In particular, the iron Kα emission line, that is the strongestobserved and is strongly affected by relativistic effects, is an interesting probe.

Many authors have recently investigated the X-ray spectral observables associatedto non-Kerr spacetimes [1720]. The same distinction presented above for the testsof shadows can be made between parametric studies [1721–1723], specific alternativetheoretical frameworks [1707, 1722, 1724–1726], and alternative objects within classicalGR [1727, 1728].

Although differences with respect to Kerr are commonly found in variousframeworks, these are generally degenerate with other parameters, such as mass, spinand inclination of the source. Moreover, the precise shape of the iron line depends onthe subtle radiative transfer in the accretion disk, which is ignored in theoretical studiesthat generally favor simple analytic models. This is a great advantage of the shadowmethod, which is probably the EM probe least affected by astrophysical systematics.

QPOs Quasi-periodic oscillations (QPOs) are narrow peaks in the power spectra,routinely observed in the X-ray light curves of binaries. QPOs are of the order ofKeplerian frequencies in the innermost regions of the accretion flow [1729].

A series of recent works were devoted to studying the QPO observables associated toalternative compact objects, be it in the context of parametric spacetimes [1730, 1731],alternative theoretical frameworks [1707, 1732, 1733], or alternative objects withinGR [1734]. Although the QPO frequencies can be measured with great accuracy, QPOdiagnostics suffer from the same limitations as the X-ray spectrum: degeneracies andastrophysical uncertainties. The non-Kerr parameters are often degenerate with theobject’s spin. Moreover, it is currently not even clear what the correct model for QPOsis.

Pulsar-BH systems High-precision timing observations of radio pulsars provide verysensitive probes of the spacetime in the vicinity of compact objects. Indeed, the firstevidence for the existence of gravitational waves were obtained from observations ofbinary pulsars [1735]; light propagation in strong gravitational effects can be preciselytested with Shapiro delay experiments [1736]; relativistic spin-precession can be studiedby examining pulse shape and polarisation properties of pulsars [1737, 1738]. The verysame techniques can be applied for pulsars to be found around BHs [1739].


While shadows, accretion-disk spectra, and QPOs are probing the near field of theBH, i.e. on a scale of a few Schwarzschild radii, it is not expected to find a pulsarthat close to a BH. The lifetime of such a pulsar-BH system due to gravitational wavedamping is very small, which makes a discovery of such a system extremely unlikely.This is also true for a pulsar around Sgr A∗, although observational evidence stills pointtowards an observable pulsar population in the Galactic Centre[1740]. Consequently, apulsar-BH system, once discovered, is expected to provide a far field test, i.e. a test ofonly the leading multipole moments of the BH spacetime, in particular mass M•, spinS• and — for IMBHs and Sgr A∗ — the quadrupole moment Q• [1741–1743]. On theone hand, the measurement of M•, S• and Q• can be used to test the Kerr hypothesis.On the other hand, a pulsar can provide a complementary test to the near field test,and break potential degeneracies of, for instance, a test based on the shadow of theBH[1702, 1717, 1744].

A pulsar in a sufficiently tight orbit (period . few days) about a stellar-mass BHis expected to show a measurable amount of orbital shrinkage due to the emission ofgravitational waves. Alternatives to GR generally show the existence of “gravitationalcharges” associated with additional long-range fields, like e.g. the scalar field in scalar-tensor theories. Any asymmetry in such “gravitational charge” generally leads to theemission of dipolar radiation, which is a particular strong source of energy loss, as itappears at the 1.5 post-Newtonian level in the equations of motion. Of particular interesthere are theories that give rise to extra gravitational charges only for BHs and thereforeevade any present binary pulsar tests. Certain shift-symmetric Horndeski theories knownto have such properties [1412, 1414, 1415, 1428], where a star that collapses to a BHsuddenly acquires a scalar charge in a nontrivial manner [1588, 1589] (cf. Sections 2.2and 4.2). Based on mock timing data, Liu et al. [1742] have demonstrated the capabilityof utilizing a suitable pulsar-BH system to put tight constraints on any form of dipolarradiation.

Finally, there are interesting considerations how a pulsar-BH system could be usedto constrain quantum effects related to BHs. For instance, there could be a change inthe orbital period caused by the mass loss of an enhanced evaporation of the BH, forinstance due to an extra spatial dimension. An absence of any such change in the timingdata of the pulsar would lead to constraints on the effective length scale of the extraspatial dimension [1745]. If quantum fluctuations of the spacetime geometry outside aBH do occur on macroscopic scales, the radio signals of a pulsar viewed in an (nearly)edge-on orbit around a BH could be used to look for such metric fluctuations. Suchfluctuations are expected to modify the propagation of the radio signals and thereforelead to characteristic residuals in the timing data [1746].

Given the prospects and scientific rewards promised by PSR-BH systems, searchesare on-going to discover these elusive objects. Pulsars orbiting stellar-mass BHs areexpected to be found in or near the Galactic plane. Since binary evolution requiressuch system to survive two supernova explosions, this implies a low systemic velocity,placing it close to its original birth place. On-going deep Galactic plane surveys, like


those as part in the High Resolution Universe Survey [1747] or upcoming surveys usingthe MeerKAT or FAST telescopes, clearly have the potential to uncover such systems.Looking at regions of high stellar density, one can expect even to find millisecond pulsarsaround BHs due to binary exchange interactions, making globular clusters [1748] andthe Galactic Centre [1749] prime targets for current and future surveys. As discussed,finding a pulsar orbiting Sgr A* would be particularly rewarding. Past surveys arelikely to have been limited by sensitivity and scattering effects due to the turbulentinterstellar medium although the discovery of a radio magnetar in the Galactic Center[1750] indicates that the situation may be more complicated than anticipated [1751].Searches with sensitive high-frequency telescopes will ultimate provide the answer [1752].Meanwhile, sensitive timing observations of pulsars in Globular cluster can also probethe proposed existence of Intermediate Mass Black Holes (IMBHs) in the cluster centresby sensing how the pulsars “fall” in the cluster potential. In some clusters, prominentclaims [1753] can be safely refuted [1754], while in other clusters IMBHs may still exist inthe centre [1755, 1756] or their potential mass can at least be constrained meaningfully[1757].

In summary, current and future radio pulsars observations have the potential tostudy BH properties over a wide mass range, from stellar-mass to super-massive BHsproviding important complementary observations presented in this chapter;

4.4.2. GW diagnostics In contrast to EM probes, GW tests are able to probealso the interior of compact objects and are much less affected by the astrophysicalenvironment [1532]. Thus, they are best suited to constraining all classes of ECOs.

GW tests based on the ringdown phase The remnant of a binary merger is a highlydistorted object that approaches a stationary configuration by emitting GWs during the“ringdown” phase. If the remnant is a BH in GR, the ringdown can be modeled as asuperposition of damped sinusoids described by the quasinormal modes (QNMs) of theBH [1122, 1516, 1517] (see Sec. 3.3). If the remnant is an ECO, the ringdown signal isdifferent:• For UCOs, the ringdown signal can be qualitatively similar to that of a BH, but

the QNMs are different from their Kerr counterpart [1548–1550]. The rates ofbinary mergers that allow for QNM spectroscopic tests depends on the astrophysicalmodels of BH population [1523]. The estimated rates are lower than 1/yr forcurrent detectors even at design sensitivity. On the other hand, rates are higher forthird-generation, Earth-based detectors and range between a few to 100 eventsper year for LISA, depending on the astrophysical model [1523]. Even if theringdown frequencies of a single source are hard to measure with current detectors,coherent mode stacking methods using multiple sources may enhance the ability ofaLIGO/aVirgo to perform BH spectroscopy [1546]. Such procedure is dependentupon careful control of the dependence of ringdown in alternative theories on theparameters of the system (mass, spin, etc).


• For ClePhOs, the prompt post-merger ringdown signal is identical to that of a BH,because it excited at the light ring [1551, 1553]. However, ClePhOs genericallysupport quasi-bound trapped modes [1758, 1759] which produce a modulated trainof pulses at late times. The frequency and damping time of this sequence ofpulses is described by the QNMs of the ClePhO (which are usually completelydifferent from those of the corresponding BH) [1760, 1761]. These modes weredubbed “GW echoes” and appear after a delay time that, in many models, scalesas τecho ∼ M log(1 − 2M/R) [1552]. Such logarithmic dependence is key to allowfor tests of Planckian corrections at the horizon scale [1552, 1553, 1762]. Modelsof ultracompact stars provide GW echoes with a different scaling [1665, 1763], thelatter being a possible smoking gun of exotic state of matter in the merger remnant.• In addition to gravitational modes, matter modes can be excited in ECOs [1506]. So

far, this problem has been studied only for boson stars [1193, 1196] and it is unclearwhether matter QNMs would be highly redshifted for more compact ECOs [1665].

GW tests based on the inspiral phase The nature of the binary components has abearing also on the GW inspiral phase, chiefly through three effects:

• Multipolar structure. As a by-product of the BH uniqueness and no-hairtheorems [1521], the mass and current multipole moments (M`, S`) of anystationary, isolated BH can be written only in terms of mass M ≡ M0 andspin χ ≡ S1/M

2. The quadrupole moment of the binary components entersthe GW phase at 2PN relative order, whereas higher multipoles enter at higherPN order [915]. The multipole moments of an ECO are generically different,e.g. MECO

2 = MKerr2 + δq(χ,M/R)M3, and it is therefore possible to constrain

the dimensionless deviation δq by measuring the 2PN coefficient of the inspiralwaveform. This was recently used to constrain O(χ2) parametrized deviationsin δq [1323]. It should be mentioned that ECOs will generically display higher-order spin corrections in δq and that – at least for the known models of rotatingultracompact objects [1764–1766] – the multipole moments approach those of aKerr BH in the high-compactness limit. Moreover, the quadrupole PN correction isdegenerate with the spin-spin coupling. Such degeneracy can be broken using theI-Love-Q relations [1767, 1768] for ECOs, as computed for instance in the case ofgravastars [1764, 1765].• Tidal heating. In a binary coalescence the inspiral proceeds through energy and

angular momentum loss in the form of GWs emitted to infinity. If the binarycomponents are BHs, a fraction of the energy loss is due to absorption at thehorizon [1769–1774]. This effect introduces a 2.5PN × log v correction to the GWphase of spinning binaries, where v is the orbital velocity. The sign of this correctiondepends on the spin [1770–1772], since highly spinning BHs can amplify radiationdue to superradiance [1156]. In the absence of a horizon, GWs are not expectedto be absorbed significantly, and tidal heating is negligible [1696, 1775]. Highly


spinning supermassive binaries detectable by LISA will provide a golden test ofthis effect [1775].• Tidal deformability. The gravitational field of the companion induces tidal

deformations, which produce an effect at 5PN relative order, proportional to the so-called tidal Love numbers of the object [918]. Remarkably, the tidal Love numbersof a BH are identically zero for static BHs [1776–1779], and for spinning BHsto first order in the spin [1780–1782], and to second order in the axisymmetricperturbations [1781]. On the other hand, the tidal Love numbers of ECOs aresmall but finite [1764, 1765, 1783, 1784]. Thus, the nature of ECOs can be probedby measuring the tidal deformability, similarly to what is done to infer the nuclearequation of state in NSBs [21, 1785, 1786]. Analysis of the LIGO data shows thatinteresting bounds on the tidal deformability can be imposed already, to the levelthat some boson star models (approximated through a polytropic fluid) can beexcluded [1787]. The tidal Love numbers of a ClePhO vanish logarithmically in theBH limit [1784], providing a way to probe horizon scales. For Planckian correctionsnear the horizon, the tidal Love numbers are about 4 orders of magnitude smallerthan those of a NS. It is therefore expected that current and future ground-baseddetectors will not be sensitive enough to detect such small effect, while LISAmight be able to measure the tidal deformability of highly-spinning supermassivebinaries [1775].

Finally, it is possible that ECOs’ low-frequency modes are excited during theinspiral, leading to resonances in the emitted GW flux [1549, 1788, 1789]. Low-frequencymodes certainly arise in the gravitational sector, as we discussed already. In addition,fluid modes at low frequency can also be excited, although this issue is poorly studied.

Challenges in modeling ECO coalescence waveforms With the notable exception ofboson stars [1678], very little is known about the dynamical formation of isolated ECOsor about their coalescence. While the early inspiral and post-merger phases can bemodelled within a PN expansion and perturbation theory, respectively, searches forECO coalescence require a full inspiral-merger-ringdown waveform. Some combinationof numerical and semianalytical techniques – analog to what is done to model preciselythe waveform of BH binaries [1101, 1111, 1297] – is missing.

Concerning the post-merger ringdown part alone, it is important to developaccurate templates of GW echoes. While considerable progress has been recentlydone [1760, 1790–1796], a complete template which is both accurate and practicalis missing. This is crucial to improve current searches for echoes in LIGO/Virgodata [1762, 1797–1803]. Model-independent burst searches have recently been reported,and will be instrumental in the absence of compelling models [1804].

5. The dark matter connectionContributors: G. Bertone, D. Blas, R. Brito, C. Burrage, V. Cardoso, L. Urena-


López

The nature and properties of DM and dark energy in the Universe are among theoutstanding open issues of modern Cosmology. They seem to be the responsible agentsfor the formation of large scale structure and the present accelerated phase of the cosmicexpansion. Quite surprisingly, there is a concordance model that fits all available set ofobservations at hand. This model is dubbed ΛCDM because it assumes that the mainmatter components at late times are in the form of a cosmological constant for the darkenergy [1805, 1806] and a pressureless component known as cold DM [1807]. These twoassumptions, together with the theoretical basis for GR, make up a consistent physicalview of the Universe [639].

The nature of the missing mass in the Universe has proven difficult to determine,because it interacts very feebly with ordinary matter. Very little is known aboutthe fundamental nature of DM, and models range from ultralight bosons with masses∼ 10−22 eV to BHs with masses of order 10M. Looking for matter with unknownproperties is extremely challenging, and explains to a good extent why DM has neverbeen directly detected in any experiment so far. However, the equivalence principleupon which GR stands – tested to remarkable accuracy with known matter – offersa solid starting point. All forms of energy gravitate and fall similarly in an externalgravitational field. Thus, gravitational physics can help us unlock the mystery of DM:even if blind to other interactions, it should still “see” gravity. The feebleness with whichDM interacts with baryonic matter, along with its small density in local astrophysicalenvironments poses the question of how to look for DM signals with GWs.

5.1. BHs as DM

In light of the LIGO discoveries there has been a revival of interest in thepossibility that DM could be composed of BHs with masses in the range 1 − 100M[227, 230, 286, 1808, 1809]. To generate enough such BHs to be DM, they would need tobe produced from the collapse of large primordial density fluctuations [285, 1810, 1811].The distribution of the BH masses that form depends on the model of inflation [1812].Such BHs can be produced with sufficiently large masses that they would not haveevaporated by the current epoch. Alternatively, DM could be composed of ultracompacthorizonless objects for which Hawking radiation is suppressed [1813]. Different formationscenarios and constraints on such objects were reviewed in Chapter I, Section 6.

If all of the DM is composed of such heavy compact objects a key signature isthe frequency of microlensing events [1814]. Microlensing is the amplification, for ashort period of time, of the light from a star when a compact object passes close tothe line of sight between us and the star. How frequent and how strong these eventswill be, depends on the distribution of BH masses [1815–1817]. It has been claimedthat DM composed entirely of primordial BHs in this mass range is excluded entirely bymicrolensing [285]; however, such study assumed that the BH mass distribution was adelta function. If the mass distribution is broadened the tension with the microlensing


data weakens [319, 320, 1818], although whether realistic models can be compatible withthe data remains a subject of debate [157]. Further observational signatures include thedynamical heating of dwarf galaxies, through two body interactions between BHs andstars [228, 316], electromagnetic signatures if regions of high BH density also have highgas densities (such as the center of the galaxy) [318, 1819], constraints from the CMBdue to energy injection into the plasma in the early universe [312, 314] and from the(absence of) a stochastic background of GWs [1809].

At the very least primordial BHs in the LIGO mass range can be a component ofthe DM in our universe. Future GW observations determining the mass and spatialdistribution of BHs in our galaxy will be key to testing this hypothesis.

5.2. The DM Zoo and GWs

Despite the accumulation of evidence for the existence of DM, the determinationof its fundamental properties remains elusive. The observations related to GWs maychange this. GWs and their progenitors ‘live’ in a DM environment to which they aresensitive (at least gravitationally though other interactions may occur). Furthermore,the DM sector may include new particles and forces that can change the nature anddynamics of the sources.

Before making any classification of DM candidates, we remind that the preferredmodels are those that: i) explain all the DM, ii) include a natural generation mechanism,iii) can be added to the SM to solve other (astrophysical or fundamental) puzzles, iv)can be tested beyond the common DM gravitational effects. None of these properties isreally necessary, but their advantages are clear.

A first distinction one can make when classifying DM models is whether the stateof DM is a distribution of fundamental particles or of compact objects. The masses ofcompact objects are determined by astrophysical evolution from initial overdensities ina matter field. As a consequence, the range and distribution of masses is quite broad.These models and their bounds are discussed in Sec. 5.1 (see also Sec. 3 for more detailsabout astrophysical BHs).

Fundamental particles have fixed mass, which constrains the mass to be mχ .MP

to treat gravity within an EFT formalism (see [1820] for a recent study of the toprange). Furthermore, there is no candidate within the SM for these fundamentalparticles. Particles with typical energy scales in the electroweak range (WIMPs) havebeen preferred in the past because they satisfy the four points above [283]. Otherpopular candidates include axions and sterile neutrinos (see e.g. Ref. [284]).

The first classification of particle models is based on the DM spin. FermionicDM can only exist with masses above∗ mχ & keV. This is because the phase-spaceavailable for fermions in virial equilibrium in dwarf spheroidals is limited [1821]. Thelimit mχ & keV always applies for models where the DM was generated thermally. In

∗ All the limits of this section apply to candidates that represent all the DM. The bounds are less stringentfor fractional components.


this case, the distribution of DM particles is too ‘hot’ for lighter masses and there isa suppression of the growth of cosmological structures at small scales, in tension withobservations [1822]. Other ‘out-of-equilibrium’ production mechanisms allow for ’cold’distributions for all possible masses, see e.g. [1823] and references therein. The extremesituation are models where a bosonic candidate is generated ‘at rest’ as happens in thecases of misalignment or inflationary generation, see [1824]. The fundamental limitationis then mχ & 10−22 eV, which comes from the wiping of structure at scales smaller thanthe de Broglie wavelength of the candidate. This scale should allow for the existence ofdwarf spheroidals. Ultra-light candidates incorporate the existence of oscillating modeswith coherence times long enough to generate new phenomena as compared to standardcandidates (see below).

The complexity of the DM sector is also unknown. Models that incorporate naturalcandidates for DM (e.g. supersymmetric models or axion models) provide concreteproposals, but one should keep an open mind when thinking about DM phenomenology.The existence of a coupling DM-SM beyond the gravitational one may mean an extrainteraction of gravitating bodies among themselves or with the DM background. Thiscan modify the orbital properties and open new emission channels, if the DM candidateis light enough. Besides, different DM models may admit different DM distributions,depending on their production mechanism and type of interactions. The common featureis the existence of spherical DM halos, but more complex structures (mini-halos, DMdisks, breathing modes,....) may occur.

Finally, the distribution of DM close to the galactic center is also quiteuncertain [1825]. This is because baryonic effects are more important. Still, one may beable to distinguish certain features, as the existence of huge overdensities, or solitons,characteristic of some DM models [1826]. For instance, the prediction of solitons incenter of galaxies for masses m ∼ (10−22÷ 10−21) eV has been recently used in [1827] toclaim that this mass range is in tension with data.

5.3. The cosmological perspective on DM: scalar field dark matter

Requiring a consistent cosmology limits some of the DM candidates, if they areto solve both the dark energy and DM puzzles. An intense experimental search in thelast years has successfully imposed stringent limits on the interactions between DMand “ordinary” matter [1828–1830]. The Weakly Interacting Massive Particle (WIMP)hypothesis [1831], while appealing, is under extreme pressure. Thus, alternative modelsneed to be seriously considered [1832]. Among the vast number of possibilities, scalarfields have become increasingly tempting [1806, 1833]. Scalar field DM is a model wherethe properties of the DM can be represented by a relativistic scalar field φ endowedwith an appropriate scalar potential V (φ) [1824, 1834–1837]. In the relativistic regime,the equation of motion for the scalar is the Klein-Gordon equation ∂µ∂µφ − ∂φV = 0,whereas the non-relativistic regime leads to a Schrödinger-type equation for the wavefunction. At the quantum level, the scalar represents the mean field value of a coherent


state of boson particles with a large occupation number.Although many possibilities exist for the potential V (φ) in DM models, it should

possess a minimum at some critical value φc around which we can define a mass scale mfor the boson particle, m ≡ ∂2

φV (φc). The simplest possibility is the parabolic potentialV (φ) = (1/2)m2φ2 (with φc = 0), but higher order terms could play a role at higherenergies. One representative example of the scalar field hypothesis is the axion fromthe Peccei-Quinn mechanism to solve the strong CP problem, for which the potentialhas the trigonometric form V (φ) = m2f 2[1 − cos(φ/f)], where f is the so-called decayconstant of the axion particle [1824]. In contrast to the parabolic potential, the axionpotential is periodic, has an upper energy limit V (φ) ≤ m2f 2 and includes contributionsfrom higher-order terms φ4, φ6, . . . [1838, 1839]. For simplicity, we present below asummary of results obtained for the parabolic potential and their constraints arisingfrom cosmological and astrophysical observations.

5.3.1. Cosmological background dynamics Because of the resemblance of the KGequation to that of the harmonic oscillator, there are two main stages in the evolutionof the scalar field: a damped phase with the Hubble constant H m during whichφ ' const., and a stage of rapid field oscillations around the minimum of the potentialthat corresponds to H m. A piece-wise solution for the two regimes can be envisagedfrom semi-analytical studies, but the real challenge arises in numerical simulations forwhich one desires a continuum solution for all the field variables. The two stages canbe put together smoothly in numerical simulations [1840]. The evolution of the energydensity ρφ = (1/2)φ2 + (1/2)m2φ2 should transit from ρφ = const. to ρφ ∝ a−3 (theexpected one for a DM component) before the time of radiation-matter equality toobtain the correct amount of DM at the present time. A lower bound on the bosonmass m ≥ 10−26eV arises from such a requirement [1840].

5.3.2. Cosmological linear perturbations For this to be a successful model, the scalarfield DM must allow the formation of structure. Generically, it is found that smalldensity fluctuations δρ/ρ grow (decay) for wavelengths k2 < 2a2mH (k2 > 2a2mH),which indicates the existence of an effective time-dependent Jeans wavenumber definedby k2

J = 2a2mH, where a is the scale factor. The most noticeable aspect about thegrowth of linear perturbations is that the power spectrum has the same amplitude asthat of CDM for large enough scales (ie k < kJ), and that a sharp-cut-off appears forsmall enough scales (ie k > kJ). The straightforward interpretation is that the SFDMmodels naturally predicts a dearth of small scale structure in the Universe, and thatsuch dearth is directly related to the boson mass. This time the lower bound on theboson mass is somehow improved to m ≥ 10−24eV [1840, 1841], but one is limited bythe non-linear effects on smaller scales to impose a better constraint.

5.3.3. Cosmological non-linear perturbations Although N -body simulations have beenadapted to the field description required for SFDM, they cannot capture the full field


dynamics and the best option remains to solve directly the so-called Schrödinger-Poissonsystem. Some studies suggest that the non-linear process of structure formation proceedsas in CDM for large enough scales [1826, 1842, 1843].

The gravitationally bound objects that one could identify with DM galaxy halosall have a common structure: a central soliton surrounded by a NFW-like envelopecreated by the interference of the Schrödinger wave functions [1826, 1844]. In termsof the standard nomenclature, the scalar-field DM model belongs to the so-called non-cusp, or cored, types of DM models (towards which evidence is marginally pointing).Simulations suggest a close relationship between the soliton Ms and the total halomass Mh of the form Ms = γ(a)M1/3

h , so that massive halos also have massive centralsolitons [1826, 1842, 1844]. Here γ(a) is a time-dependent coefficient of proportionality.

These results then predict a minimum halo mass Mh,min = γ3/2(1) '(m/10−22 eV)−3/2 107M when the galaxy halo is just the central soliton. But the solitonscale radius rs is related to its mass via Msrs ' 2(mPl/m)2, where mPl = G−1/2 isthe Planck mass. For the aforementioned values of the soliton mass, rs = 300 pc ifMs = 107M, in remarkable agreement with observations [1845] (see, however, [1846]).Furthermore, the previous relation between the soliton mass and the host halo masscan be used to predict the presence of a second peak in the velocity of rotation curves,closer to the galactic center [1827]. This prediction is in tension with data for massesin the range m ∼ (10−22 ÷ 10−21) eV [1827].

Finally, non-linear results also constrain the scalar mass through comparisonwith observations inferred from the Lyman-α forest [1847]. These constraints requirededicated N -body simulations with gas and stars, so that the flux of quasars canbe calculated. The constraints are obtained indirectly and subjected to uncertaintiesarising from our limited knowledge on the formation of realistic galaxies. In a firstapproximation we can estimate the power spectrum of the transmitted flux of theLyman-α forest at linear order, and obtain a lower limit on the boson massm > 10−21eVif SFDM is to fulfill the DM budget completely [1848].

In conclusion, scalar-field models of dark energy can account for the whole of DMas well, provided the scalar has a mass of order 10−21 eV. This model explains also theseemingly cored density profile in the central parts of the galaxies.

5.4. The DM environment

In the following we provide a (hopefully agnostic) view on how different DM modelsmay affect GW emission by compact objects. The environment in which compact objectslive – be it the interstellar medium plasma, accretion disks or DM – can influence theGW signal in different ways:

• By modifying the structure of the compact objects themselves. Accretion disks, forexample, alter the spacetime multipolar structure relative to the standard Kerrgeometry. Accretion disks are also known to limit the parameter space of KerrBHs, preventing them from spinning beyond cJ/(GM2) ∼ 0.998 [1103]. Likewise,


if DM behaves as heavy particles, its effects parallel that of baryonic matter eitherin accretion disks or in the interstellar medium.In models where a particle description is insufficient (for example, if DM assumesthe form of fundamental light bosons), Kerr BHs can either be unstable, or notbe an equilibrium solution of the field equations at all [1407, 1519, 1609] (seealso discussion in Section 4). Some of the simplest forms of axionic DM havea more mild effect instead, contributing to the spinning down of astrophysicalBHs [1147, 1849, 1850].Independently of the nature of DM, compact stars evolving in DM-richenvironments may accrete a significant amount of DM: DM is captured by thestar due to gravitational deflection and a non-vanishing cross-section for collisionwith the star material [1167, 1851–1854]. The DM material eventually thermalizeswith the star, and accumulates inside a finite-size core [1167, 1852, 1853, 1855].• By altering the way that GWs are generated. The different equilibrium configuration

of the compact objects, together with the possible self-gravity effects of DM anddifferent structure of the field equations gives rise, generically, to a different GWoutput from that predicted in vacuum GR.• The environment – interstellar dust, a cosmological constant or DM – must of

course affect also the way that GWs and electromagnetic waves propagate. It is acommon-day experience that light can be significantly affected by even low-densitymatter. For example, low-frequency light in a plasma travels at the group velocityvg = c

√1− ω2

p/ω2, with ωp the plasma frequency ω2

p = n0q2

4πε0 m . Here, n0 is theparticle number density, q the electron charge and m its mass. Thus, light isdelayed with respect to GWs, by an amount δt directly proportional to the totaldistance they both traversed

δt = dρ0q2

8πε0 cm2ω2

= 6.7 ρ0

1 GeV/cm3

(6GHzω

)2

days , (39)

where in the last equality we substituted numbers for the very latest observation ofGWs with an electromagnetic counterpart [22]. These numbers could be interesting:given the observed time delay of several days for radio waves, one may getconstraints for models where DM consists of mili-charged matter (i.e., models whereq and m may be a fraction of the electron charge and mass).Cold DM affects the propagation of GWs by inducing a small frequency dependentmodification of the propagation speed [1856]. Furthermore, the effect of viscousfluids or oscillators on the passage of GWs are discussed in Refs. [1857, 1858].These calculations indicate that the effect may be too small to be measurable inthe near future.

These different effects have, so far, been explored and quantified in the context ofa few possible mechanisms where DM plays a role.


5.5. Accretion and gravitational drag

Figure 12. Depiction of a BBH evolving in a possible interstellar or DM environment.Each individual BH accretes and exherts a gravitational pull on the surroundingmatter. Both effects contribute to decelerate the BHs, leading to a faster inspiral.Credit: Ana Sousa.

The formation and growth of compact objects may lead to large overdensities of DMaround them [1859–1861]. Whatever the nature and amount of DM surrounding them, abinary of two BHs or compact stars evolving in a DM-rich environment will be subjectedto at least two effects, depicted in Fig. 12. As it moves through the medium it accretesthe surrounding material, but also exherts a gravitational pull (known as “gravitationaldrag”) on all the medium, which affects the inspiral dynamics. To quantify these effects,it is important to know how DM behaves. Collisionless DM causes, generically, agravitational drag different from that of normal fluids [1549, 1789]. The gravitationaldrag caused by DM which is coherent on large scales may be suppressed [1836], butfurther work is necessary to understand this quantitatively.

The phase evolution of a pair of BHs, taking gravitational radiation, accretion anddrag as well as the DM self-gravity was studied recently [1532, 1549, 1789, 1862–1864].


These estimates are all Newtonian, and the results depend crucially on the local DMdensity. Perhaps more importantly, back-reaction on the DM distribution was not takeninto account in a systematic manner. The only studies available including backreaction,look only at the very last orbits of the binary, and are unlikely to capture well the fullphysics of DM dispersal [1107].

5.6. The merger of two dark stars

Gravitational drag and accretion have a cumulative effect: even though theirmagnitude is small, the effect piles up over a large number of orbits. By contrast,drag and accretion have a tiny effect during merger. The merger signal carries, mostly,an imprint of the configuration of the colliding objects. For BHs, the no-hair resultsstrongly suggest that the colliding objects belong to the Kerr family, and it seemshopeless to find any DM imprints here. However, the no-hair results can be by-passed,as they assume stationarity of the spacetime or certain matter properties, as explainedin Sec. 4 [1407, 1519, 1609]. Detailed investigations of BHs other than Kerr are stillto be done. Smoking-gun effects arising from the merger of such objects are, likewise,unknown.

As we discussed, compact stars can grow DM cores at their center. It is conceivablethat the DM core might imprint a signature on different phases of the coalescence oftwo of these objects. The effects of tidal deformations within the gravitational wavesignal produced by binary DM stars have been investigated in [1865]. Moreover, theimprint on the post-merger spectrum, so far only for analog-mechanical models, hasbeen studied in [1866].

5.7. Non-perturbative effects: superradiance and spin-down.

Spinning BHs are huge energy reservoirs from which all rotational energy canultimately be extracted. In particular, due to the absence of a Pauli exclusion principlefor bosons, bosonic waves can effectively extract energy from spinning BHs through aprocess known as rotational superradiance [1156]. The superradiant energy extractionmight be used to rule out (or detect) ultralight bosons that might be a significantcomponent of DM, such as the QCD axion or axion-like particles predicted by thestring axiverse scenario [1147], dark photons [1529, 1867] and even massive spin-2fields [1560]. For boson masses in the range 10−21 – 10−11 eV, their Compton wavelengthis of the order of the horizon of typical astrophysical BHs, the gravitational couplingof these two objects is strongest, and long-lived quasi-bound states of bosonic particlesaround BHs can form. For rotating BHs these states are typically superradiant, thusbecoming an effective tap of rotational energy. The extracted energy condenses as abosonic cloud around the BH producing “gravitational atoms” [1147, 1166, 1849]. Theevolution of these systems leads to several observational channels. For complex bosonicfields, GW emission is suppressed and the end-state of the superradiant instabilitymight be a Kerr BH with bosonic hair [1168, 1171]. These solutions are themselves


unstable [1632] which, over the age of the Universe, would likely lead to a slowly-rotating BH surrounded by a bosonic cloud populating the most unstable modes. Onthe other hand for real bosonic fields, this cloud would disperse over long timescalesthough the emission of nearly-monochromatic GWs, either observable individually or asa very strong stochastic GW background [1850, 1868–1872]. These are appealing GWemitters with appropriate amplitude and frequency range for Earth- and space-baseddetectors. Although estimates from Ref. [1850, 1872] suggest that current LIGO datacan already be used constrain some range of scalar field masses, an analysis on real datahas not yet been performed.

These models also predict that BHs affected by the instability would spin-downin the process, and therefore the mere observation of rotating BHs can place severelimits on these particles, thereby strongly constraining the theory. In fact, observationsof spinning BHs can already be used to bound the photon [1529, 1867] and gravitonmasses [1560]. These studies demonstrate that BHs have an enormous potential astheoretical laboratories, where particle physics can be put to the test using currentobservations.

All these studies neglect the possible decay of ultralight bosonic fields into StandardModel particles which is justified given the current constraints on ultralight bosons inthis mass range (see e.g. [1824, 1873]). However, the superradiant instability leads to thegeneration of boson condensates around BHs with a very large particle number, whichcould enhance their decay through channels other than gravitational. For the couplingtypical of the QCD axion with photons, stimulated decay into photon pairs becomessignificant for axion masses above & 10−8 eV, and consequently for BHs with masses. 0.01M [1874]. If such systems exist, they would lead to extremely bright lasers withfrequencies in the radio band. For the expected mass of the QCD axion ∼ 10−5 eV, suchlasing effect would occur around primordial BHs with masses around ∼ 1024 Kg andlead to millisecond radio-bursts, which could suggest a link to the observed fast radiobursts [1874].

In addition to these effects, the coupling of light DM bosons with Standard Modelparticles may also turn NSs unstable against superradiant amplification. Rotationalsuperradiance is in fact a generic process and can happen in any rotating object, aslong as some sort of dissipation mechanism is at work [1156]. Thus, even NSs can besuperradiantly unstable against some models of DM, such as dark photons coupled tocharged matter, which can already be used to put constraints on these models [1875].

An important question that remains to be fully investigated is: if ultralight bosonsdo exist and we detect them through one or several of these channels, will we be able tostart using compact objects to do precision particle physics, in particular can one inferfundamental properties such as their mass, spin and fundamental interactions? Thiswill require a better understanding of higher-spin fields around rotating BHs.

Although some progress has been made to deal with massive vector fields usingtime-domain simulations, either in full GR [1152, 1166] or by fixing the backgroundgeometry to be a Kerr BH [1150, 1876], frequency-domain computations, which are more


appropriate to span the parameter space, have mainly been dealt with by employing asmall-spin approximation [1529, 1560, 1867], using an analytical matched asymptoticsapproach [1871] or using a field theory approach [1877], these last two being only validin the Newtonian limit. Computations of the instability rates without employing anyof these approximations have only recently started to be available [1878, 1879]. Anestimate of the stochastic background coming from vector or tensor fields has also notbeen computed. Finally, besides some some estimates on the GWs from a bosenovacollapse due to axion self-interations [1880], a detailed analysis of how self-interactionsor interactions with other fields could affect the GW emission and its detectability isstill missing.

5.8. Pulsar timing

The gravitational drag from particle DM candidates in the orbital motion of binarypulsars has been studied in [1881, 1882]. This may be important to constrain DM modelsthat generate a thick DM disk [1882] or to study the DM density closer to the galacticcenter, if a pulsar binary is discovered there and can be timed to high precision.

DM, specially if under the form of light bosons, can also produce other peculiareffects, accessible through EM observations. In models where DM comes in the form ofan ultralight boson, the DM configuration is able to support itself through pressure,which is in turn caused by a time-dependence of the field. This time-periodicitygives rise to an oscillating gravitational potential which is able to produce tiny, butpotentially measurable, differences in the time-of-arrival signal from precise clocks, suchas pulsars [1883, 1884].

5.9. Mini-galaxies around BHs

In addition, in these theories, BHs can grow (permanent or long-lived) time-oscillating, non-axisymmetric bosonic structures [1156]. Planets or stars in the vicinityof these BH are subjected to a periodic forcing, leading to Lindblad and co-rotationresonances. This phenomenon is akin to the forcing that spiral-armed galaxies exherton its planets and stars, and it is conceivable that the same type of pattern appears, onsmaller-scales, around supermassive hairy BHs [1864, 1885].

5.10. GW detectors as DM probes

Finally, there are interesting proposals to use the GW detectors themselves asprobes of DM, when used as extremely sensitive accelerometers [1886–1888].


PostscriptGravity sits at the heart of some of the most important open problems in

astrophysics, cosmology and fundamental physics, making it a subject of stronginterdisciplinarity. Black holes are, in some ways, the “atoms” of gravity, the “simplest”astrophysical objects, yet they harbour some of the most remarkable predictions of GR,including that of its own ultimate failure. Gravitational-wave astronomy will allowus to test models of BH formation, growth and evolution, as well as models of GWgeneration and propagation. It will provide evidence for event horizons and ergoregions,test GR itself and may reveal the existence of new fundamental fields. The synthesisof these results has the potential to shed light on some of the most enigmatic issues incontemporary physics. This write-up summarized our present knowledge and key openchallenges. We hope that it can serve as a guide for the exciting road ahead.

Acknowledgements

This article is based upon work from COST Action CA16104 “GWverse”, supportedby COST (European Cooperation in Science and Technology). We would like to thankWalter del Pozzo for useful comments. A.A. acknowledges partial support from thePolish National Science Center (NCN) through the grant UMO-2016/23/B/ST9/02732and is currently supported by the Carl Tryggers Foundation through the grant CTS17:113. L.B. acknowledges support from STFC through Grant No. ST/R00045X/1.V.C. acknowledges financial support provided under the European Union’s H2020 ERCConsolidator Grant “Matter and strong-field gravity: New frontiers in Einstein’s theory”grant agreement no. MaGRaTh–646597. T.P.S. acknowledges partial support fromScience and Technology Facilities Council Consolidated Grant ST/P000703/1. K. B.acknowledges support from the Polish National Science Center (NCN) grant: SonataBis 2 (DEC-2012/07/E/ST9/01360). E. B. acknowledges financial support from projects176003 and 176001 by the Ministry of Education and Science of the Republic of Serbia.T. B. was supported by the TEAM/2016-3/19 grant from FNP. P. C. acknowledgessupport from the Austrian Research Fund (FWF), Project P 29517-N16, and by thePolish National Center of Science (NCN) under grant 2016/21/B/ST1/00940. B. K.acknowledges support from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme ERC-2014-STG undergrant agreement No 638435 (GalNUC) and from the Hungarian National Research,Development, and Innovation Office grant NKFIH KH-125675. G. N. is supportedby the Swiss National Science Foundation (SNF) under grant 200020-168988. P. P.acknowledges financial support provided under the European Union’s H2020 ERC,Starting Grant agreement no. DarkGRA–757480. U. S. acknowledges H2020-MSCA-RISE-2015 Grant No. 690904, NSF Grant No. PHY-090003 and PRACE Tier-0 GrantNo. PPFPWG. The Flatiron Institute is supported by the Simons Foundation. P.A. S. acknowledges support from the Ramón y Cajal Programme of the Ministry ofEconomy, Industry and Competitiveness of Spain. E. B. supported by NSF Grants


No. PHY-1607130 and AST-1716715. K. B., A. C., A. G., N. I., T. P. and G. Z.acknowledge financial support from the Slovenian Research Agency. S. B. acknowledgessupport by the EU H2020 under ERC Starting Grant, no. BinGraSp-714626. P. C-D.is supported by the Spanish MINECO (grants AYA2015-66899- C2-1-P and RYC-2015-19074) and the Generalitat Valenciana (PROMETEOII-2014-069). D. G. is supportedby NASA through Einstein Postdoctoral Fellowship Grant No. PF6–170152 by theChandra X-ray Center, operated by the Smithsonian Astrophysical Observatory forNASA under Contract NAS8–03060. R. E. acknowledges financial support provided bythe Scientific and Technical Research Council of Turkey (TUBITAK) under the grantno. 117F296. J. A. F. is supported by the Spanish MINECO (AYA2015-66899-C2-1-P),by the Generalitat Valenciana (PROMETEOII-2014-069), and by the H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. F. M. R. is supported by the Scientific andTechnological Research Council of Turkey (TÜBİTAK) project 117F295. I. R. wassupported by the POLONEZ programme of the National Science Centre of Polandwhich has received funding from the European Union‘s Horizon 2020 research andinnovation programme under the Marie Skłodowska-Curie grant agreement No. 665778.A. S. thanks the Spanish Ministry of Economy, Industry and Competitiveness, theSpanish Agencia Estatal de Investigación, the Vicepresidència i Conselleria d’Innovació,Recerca i Turisme del Govern de les Illes Balears, the European Commission, theEuropean Regional Development Funds (ERDF). N. S. acknowledges support fromDAAD Germany-Greece Grant (ID 57340132) and GWAVES (pr002022) of ARIS-GRNET(Athens). H. W. acknowledges financial support by the Royal Society UKunder the University Research Fellowship UF160547-BHbeyGR and the Research GrantRGF −R1− 180073. K. Y. is supported by TÜBİTAK-117F188.

1 Mathematical Sciences, University of Southampton, Southampton SO17 1BJ,United Kingdowm2 CENTRA, Departamento de Física, Instituto Superior Técnico – IST, Universidadede Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal3 Perimeter Institute for Theoretical Physics, 31 Caroline Street North Waterloo,Ontario N2L 2Y5, Canada4 Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box9010, 6500 GL Nijmegen, The Netherlands5 GRAPPA, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, TheNetherlands6 Nikhef, Science Park, 1098 XG Amsterdam, The Netherlands7 School of Mathematical Sciences, University of Nottingham, University Park,Nottingham, NG7 2RD, UK8 School of Physics and Astronomy, University of Nottingham, University Park,Nottingham, NG7 2RD, UK9 Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, ul.Bartycka 18, 00-716 Warsaw, Poland


10 Lund Observatory, Department of Astronomy, and Theoretical Physics, LundUniversity, Box 43, SE-221 00 Lund, Sweden11 Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia12 Isaac Newton Institute of Chile, Yugoslavia branch, Serbia13 Theoretical Particle Physics and Cosmology Group, Physics Department, King’sCollege London, University of London, Strand, London WC2R 2LS, UK14 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), AmMuhlenberg 1, Potsdam-Golm, 14476, Germany15 Astronomical Observatory, University of Warsaw, Aleje Ujazdowskie 4, 00478Warsaw, Poland16 Department of Physics & Astronomy, University of Sussex, Brighton BN1 9QH,United Kingdom17 APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3,CEA/Irfu, Observatoire de Paris, Sorbonne Paris Cité, F-75205 Paris Cedex 13,France18 School of Physical Sciences and Centre for Astrophysics & Relativity, Dublin CityUniversity, Glasnevin, Dublin 9, Ireland19 Dublin Institute of Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland20 University of Vienna, Boltzmanngasse 5, A 1090 Wien, Austria21 Institut des Hautes Études Scientifiques, Le Bois-Marie, 35 route de Chartres,91440 Bures-sur-Yvette, France22 Università degli Studi di Milano-Bicocca, Dip. di Fisica “G. Occhialini”, Piazzadella Scienza 3, I-20126 Milano, Italy23 INFN – Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy24 Dipartimento di Fisica, Università di Roma “Sapienza” & Sezione INFN Roma1,P.A. Moro 5, 00185, Roma, Italy25 School of Mathematics, University of Edinburgh, James Clerk Maxwell Building,Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK26 Instituto de Física Teórica UAM/CSIC, Universidad Autónoma de Madrid,Cantoblanco 28049 Madrid, Spain27 Department of Physics & The Oskar Klein Centre, Stockholm University,AlbaNova University Cetre, SE-106 91 Stockholm28 Institute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich,Switzerland29 SUPA, University of Glasgow, Glasgow G12 8QQ, United Kingdom30 Departamento de Física da Universidade de Aveiro and CIDMA, Campus deSantiago, 3810-183 Aveiro, Portugal31 Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem,91904, Israel32 Institute of Physics, Eötvös University, Pázmány P. s. 1/A, Budapest, 1117,Hungary33 Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn,Germany34 Jodrell Bank Centre for Astrophysics, The University of Manchester, M13 9PL,United Kingdom35 LUTH, Observatoire de Paris, PSL Research University, CNRS, Université ParisDiderot, Sorbonne Paris Cité, 92190 Meudon, France36 Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, NewYork, NY 10010 US37a Universität Bern, Institute for Theoretical Physics, Sidlerstrasse 5, CH-3012Bern, Switzerland


37b Faculty of Science and Technology, University of Stavanger, 4036 Stavanger,Norway38 Departament de Física & IAC3, Universitat de les Illes Balears and Institutd’Estudis Espacials de Catalunya, Palma de Mallorca, Baleares E-07122, Spain39 INFN, Sezione di Milano-Bicocca, gruppo collegato di Parma, Parco Area delleScienze 7/A, IT-43124 Parma, Italy40 Dipartimento di Fisica G. Occhialini, Università degli Studi di Milano-Bicocca,Piazza della Scienza 3, IT-20126 Milano, Italy41 Leiden Observatory, Leiden University, PO Box 9513 2300 RA Leiden, theNetherlands42 School of Physics and Astronomy and Institute of Gravitational Wave Astronomy,University of Birmingham, Edgbaston B15 2TT, United Kingdom43 Theoretical Astrophysics, Walter Burke Institute for Theoretical Physics,California Institute of Technology, Pasadena, California 91125, USA44 Department of Applied Mathematics and Theoretical Physics, Centre forMathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB30WA, UK45 INAF, Osservatorio Astronomico di Roma, via Frascati 33, Monte Porzio Catone,00078, Rome, Italy46 Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy47 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, D-53121Bonn, Germany48 Departamento de Física, DCI, Campus León, Universidad de Guanajuato, 37150,León, Guanajuato, México49 LESIA, Observatoire de Paris, PSL Research University, CNRS, SorbonneUniversit és, UPMC Univ. Paris 06, Univ. Paris Diderot, Sorbonne Paris Cité, 5place Jules Janssen, 92195 Meudon, France50 Institut d’Astrophysique de Paris, CNRS & Sorbonne Universités, UMR 7095, 98bis bd Arago, 75014 Paris, France51 School of Mathematics and Statistics, University College Dublin, Belfield, Dublin4, Ireland52 Department of Physics, University of Virginia, Charlottesville, Virginia 22904,USA53 Departamento de Astronomía y Astrofísica, Universitat de València, C/Dr. Moliner 50, 46100, Burjassot (Valencia), Spain54 Institute of Space Sciences (ICE, CSIC) & Institut d’Estudis Espacials deCatalunya (IEEC)at Campus UAB, Carrer de Can Magrans s/n 08193 Barcelona, Spain55 Institute of Applied Mathematics, Academy of Mathematics and Systems Science,CAS, Beijing 100190, China56 Kavli Institute for Astronomy and Astrophysics, Beijing 100871, China57 Zentrum für Astronomie der Universität Heidelberg, AstronomischesRechen-Institut, Mönchhofstr. 12-14, D-69120, Heidelberg Germany58 Department of Physics and Astronomy, Johns Hopkins University, 3400 N. CharlesStreet, Baltimore, MD 21218, USA59 Department of Mathematics, Faculty of Natural Sciences and Mathematics,University of Tuzla, Univerzitetska 4, 75000 Tuzla, Bosnia and Herzegovina60 Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, 07743,Jena, Germany61 Department of Physics and Astronomy, The University of Mississippi, University,MS 38677, USA


62 Ikerbasque and Department of Theoretical Physics, University of the BasqueCountry, UPV/EHU 48080, Bilbao Spain63 Institut für Physik, Universität Oldenburg, Postfach 2503, D-26111 Oldenburg,Germany64 DiSAT, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy65 Department of Astronomy, Petnica Science Center, P. O. B. 6, 14104 Valjevo,Serbia66 Faculty of Electrical Engineering and Computing, University of Zagreb, 10000Zagreb, Croatia67 Center for Astrophysics and Cosmology, University of Nova Gorica, Vipavska 13,5000 Nova Gorica, Slovenia68 Center for Theoretical Astrophysics and Cosmology, Institute for ComputationalScience, University of Zurich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland69 Laboratoire de Physique Théorique, CNRS, Univ. Paris-Sud, UniversitéParis-Saclay, F-91405 Orsay, France70 AIM, UMR 7158, CEA/DRF, CNRS, Université Paris Diderot, Irfu/Serviced’Astrophysique, Centre de Saclay, FR-91191 Gif-sur-Yvette, France71 Eberhard Karls University of Tuebingen, Tuebingen 72076, Germany72 Université Libre de Bruxelles, Campus Plaine CP231, 1050 Brussels, Belgium73 Departamento de Matemáticas, Universitat de València, C/ Dr. Moliner 50,E-46100 Burjassot, València, Spain74 Department of Mathematics and Applied Mathematics, University of Cape Town,Private Bag, Rondebosch, South Africa, 770075 HAS Wigner Research Centre for Physics, H-1525 Budapest, Hungary76 Department of Physics, University of Naples "Federico II", Via Cinthia, I-80126,Naples, Italy77 Istituto Nazionale di Fisica Nucleare (INFN), Sez. di Napoli, Via Cinthia 9,I-80126, Napoli, Italy78 INRNE - Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria79 Physics Department, University of Lancaster, UK80 Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig LluísCompanys 23, E-08010 Barcelona, Spain81 Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos,Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain82 Department of Physics, Izmir Institute of Technology, Izmir, Turkey83 Astrophysics, University of Oxford, DWB, Keble Road, Oxford OX1 3RH, UK84 Institute of Space Sciences and Astronomy of the University of Malta85 Observatori Astronòmic, Universitat de València, C/ Catedrático José Beltrán 2,46980, Paterna (València), Spain86 Institute for Theoretical Physics, KULeuven, Celestijnenlaan 200D, 3001 Leuven,Belgium87 Faculty of Physics, Moscow State University, 119899, Moscow, Russia88 Kazan Federal University, 420008 Kazan, Russia89 Department of Mathematics, Rhodes University, Grahamstown 6140, South Africa90 Departamento de Física, Universidad de Murcia, Murcia E-30100, Spain91 Physics Department, Ben Gurion University, Beer Sheva, Israel92 DiSAT, Università dell’Insubria, Como, Italy93 Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100Copenhagen, Denmark94 Department of Physics and Astronomy, Earlham College, Richmond, IN 47374 US


95 ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA,Dwingeloo, The Netherlands96 Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park904, 1098 XH, Amsterdam, The Netherlands97 Department of Physics, University of Zurich, Winterthurerstrasse 190, CH-8057Zurich98 Section of Astrophysics, Astronomy and Mechanics, Department of Physics,National and Kapodistrian University of Athens, 15784 Zografos, Athens, Greece99 Astrophysics Research Institute, LJMU, IC2, Liverpool Science Park, 146Brownlow Hill, Liverpool L3 5RF, UK100 Center for Applied Space Technology and Microgravity (ZARM), University ofBremen, Am Fallturm, 28159 Bremen, Germany101 Monash Centre for Astrophysics, School of Physics and Astronomy, MonashUniversity, VIC 3800, Australia102 OzGrav: The ARC Centre of Excellence for Gravitational-wave Discovery,Clayton, VIC 3800, Australia103 SISSA, International School for Advanced Studies, Via Bonomea 265, 34136Trieste, Italy104 INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy105 School of Chemistry and Physics, University of KwaZulu-Natal, WestvilleCampus, Private Bag X54001, Durban, 4000, South Africa106 NAOC-UKZN Computational Astrophysics Centre (NUCAC), University ofKwaZulu-Natal, Durban, 4000, South Africa107 Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008,China108 Consortium for Fundamental Physics, School of Mathematics and Statistics,University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, Sheffield S3 7RH,United Kingdom109 Van Swenderen Institute, University of Groningen, 9747 AG, Groningen, TheNetherlands110 Institute of Theoretical Astrophysics, University of Oslo, 0315 Oslo, Norway111 Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy ofSciences, Boul. Tsarigradsko Chausse 72, BG-1784 Sofia, Bulgaria112 Service de Physique Théorique et Mathématique, Université Libre de Bruxellesand International Solvay Institutes, Campus de la Plaine, CP 231, B-1050 Brussels,Belgium113 Department of Mathematics, Faculty of Natural Sciences and Mathematics,University of Tuzla, Univerzitetska 4, 75000 Tuzla, Bosnia and Herzegovina114 Faculty of Natural Sciences and Mathematics, University of Banjaluka, MladenaStojanovića 2, 78000 Banjaluka, Republic of Srpska, Bosnia and Herzegovina115 Department of Astronomy, Faculty of Mathematics, University of Belgrade,Studentski Trg 16, 11000 Belgrade, Serbia116 Faculty of Physics, University of Warsaw, Ludwika Pasteura 5, 02-093 Warsaw,Poland117 Wigner RCP, H-1121 Budapest, Konkoly Thege Miklós út 29-33, Hungary118 Department of Physics, Koç University, Rumelifeneri Yolu, 34450 Sariyer,Istanbul, Turkey119 Institute of Physics, Maria Curie-Skłodowska University, Pl. MariiCurie-Skłodowskiej 1, Lublin, 20-031, Poland120 Janusz Gil Institute of Astronomy, University of Zielona Gora, Licealna 9, 65-417,Zielona Gora, Poland

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